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I am stealing Charlie Brumbaugh's question about canoeing on an open lake, because for kayaking you would get different anwers, and it's important to distinguish those.

If you are kayaking on a lake and the wind suddenly starts blowing to the point where there are whitecaps, what is the best direction to head to get off the lake?

Would it be the shortest way back to the shore or would one take the direction of the wind into account?

Assume a one-person kayak with the cockpit covered.

First of all, if the wind suddenly starts blowing, realize that this is a case of bad planning.

You should never find yourself in the middle of an open lake when the weather patterns may lead to powerful winds.

But it's too late now, and you have to make a decision where to paddle to.

(This answer focuses on longer type kayaks (5m), because I've never used shorter ones for trips; they just don't 'have the mileage'. See Separatrix's answer addressing shorter boat types.)

The short answer, in contrast with an open canoe, is:

Upwind

Nothing gives you more stability than steering you nose into the wind. As long as you keep pointing into the wind, you won't have issues with balance, even when the waves roll over your deck.

A second advantage is that, how closer to the windward lake side you get, the lower the waves will be.

The longer answer is:

It depends

These are other factors to take into account, and that may make you decide differently:

• 
  Distance to shore. If upwind to the shore is much farther than other directions you may chose differently. Note that your speed will be low upwind.


• 
  Shore structure, specifically: are there exit points, can you climb on shore? If the shore you are pointing to has heavy rocks, concrete blocks, lots of reeds, or tangled bushes, it will be hard to get out of your boat.


• 
  Kayak type. If you're in a sea kayak, it will tend to lift up on an incoming wave. A race type kayak would cut straight into the oncoming waves, and they would wash over your deck completely. That's not really an issue for an experienced paddler, but you get really wet (cold?).
  
  The boat type also determines how well you will be able to diagonally cut through the waves. A sea canoe can just 'corkscrew' through the incoming waves and an experienced sea canoeist may just like that...
  
  Never try to steer parallel to the waves though; even in a sea canoe that's hard, unless the wave are really long:


• 
  Wave height and length. The larger and deeper the lake, the longer the waves will be (distance between wave tops). Long is good. OTOH Wind pushing against the waves across long stretches (a long fetch) will give larger waves.


• And the usual: experience, fatigue, but that's not very specific to this situation.

Additional notes:

• If it's a short burst of wind, like the downdraft under a rain cloud or an incoming weather front, you may choose to 'ride it out'. Steer your nose into the wind and paddle enough to stay in place (save energy). You may want to lower your arms (paddle closer to your deck) to catch less wind on the blades.


• If you are with more people, decide together. Do not split up (generally). Take care to listen to the people least secure or experienced.


• When kayaking on large lakes, take the same precautions like when you are at sea in large estuaries: go with a minimum of three people, take flares with you, use maps, plan your trip etc.

Conditions in which you can handle the boat depend on the style of boat and the paddler

• The range of what defines a "covered cockpit kayak" includes play boats from under 2 meters to over 5 meter touring and racing boats. A considerably greater variety than "open canoe" covers.




• Paddlers range from total novices to the highly experienced or internationally competitive.

At one end of these scales you're going to have serious difficulty travelling upwind at all under these conditions, at the other end of the scale you're unlikely to really notice until conditions become significantly more adverse.

Advice for each combination of boat and paddler varies. One thing you will notice is that kayaks tend to turn nose into the waves, making upwind an easier option than you might expect.

• The easiest way off for a short boat will always be downwind, these boats are designed to surf. They'll catch a wave and you can ride it to shore. They really struggle to climb waves and you won't make much progress if you turn into the wind and waves.




• Crosswind requires some skill at handling your boat in waves, but if you lift the edge towards the incoming waves and let it turn up the wave and ride down the back it's not too bad. The chances are anyone needing this advice will be in a stable enough boat that there's not a massive problem but should generally be avoided. Taking waves straight side on should also be avoided, however as already mentioned, your boat will naturally turn up into the waves as long as you keep it upright.




• Upwind could be your better bet in longer touring boats, racing boats or sea kayaks. The boat will naturally turn this way and it won't cause any difficulties with handling. However these boats can also surf, so a good paddler should be able to take a quick ride out by going downwind. Anything above the 4 meter mark is unlikely to be slowed by small white horses, smaller touring boats will get wetter, but the point of the covered cockpit is to allow you to stay out in considerably more adverse conditions than open boats can handle.

In summary: A competent kayaker in a suitable boat doesn't actually need to head off the lake at the point where white horses are starting to form, however beginners should take an exit that suits their abilities and their boat.

The plural of anecdote isn't data: As a general rule once the white horses start appearing I treat it like a resistance session and dig in. I'm usually in a polo boat which isn't designed to handle at all in big water, however I'm also not your average recreational paddler. I advise beginners to stay away from areas where the waves tend to build when the wind picks up.

As said elsewhere “It depends”—-on the paddler’s fitness and skill, the geography of the lake, and whether the equipment (boat and clothing) make capsizing inconvenient or dangerous.

Generally speaking, paddling into the wind and waves requires greater fitness and less skill (you must maintain a pace and minimze rests to make headway, but it is easy to keep the boat oriented to the wind, and the wave action will not tend to tip the boat). Paddling downwind requires less fitness but greater skill—-at best you can deliberately surf, at worst you must be able to stay upright when you unintentionally surf and then broach (turn sideways). Paddling across the wind requires both fitness (you may have to paddle one side exclusively) and skill.

The geography of the lake means not all water is of equal danger. Waves form in response to “fetch” (length of water over which wind is blowing) so you may not need to actually reach shore to get to a part of the lake away from large waves. You may even be sheltered from wind by hills etc. Closely related, the shore on which the waves land (the lee shore) may be more or less safe to land on depending on the wave height (caused by wind speed, fetch, and time) and whether it is sandy, rocky, etc. You may have to be very close to shore to see the very real danger of concussion (which may lead to drowning), spinal cord injury, fracture, sprain that can result from abruptly “dumping” waves on sand, or interacting with rocks, even with relatively small waves. It is entirely possible to paddle to a lee shore only to realize there is no safe place to land (although usually in a lake you can at least abandon the boat and swim in). It also matters where exactly will you be on land and is that a means to safety.

Finally whether the paddler is wearing a PFD and is dressed for immersion without hypothermia, and whether the boat has appropriate flotation and rescue gear (pump, paddle float) determines whether a capsize is inconvenient (you’ll float a while but even if you can’t get back in you’ll be blown to shore eventually) or fatal (you will get hypothermic, lose the ability to self-rescue and eventually to swim, and then drown). That might influence your decision to avoid or enjoy the tippier downwind run to shore.