How safe is the Boeing 737-800 in high winds?
Extremely safe. 

Firstly, your pilots will have access to much more detailed and real-time weather information than you can get. They want to get home safely as much as you do, and will not fly if it is not safe to do so.

If the wind is coming from straight ahead, there is no maximum limit, which is good as aircraft takeoff and land into the wind whenever possible. If the wind is coming from the side - known as a crosswind - the limit in a 737 is about 36 knots/19 mps (the specific number can vary but that's a guide). The reason for this limit is to avoid the winnd blowing the plane off the side of the runway. So if the wind is that strong, hopefully the runway faces into wind, otherwise your pilot may have to delay the takeoff until it calms a bit. 

Once the aircraft lifts off the ground, the wind speed is irrelevant (except for navigation purposes). Imagine a fast flowing river. Placing your canoe into the river when it is at such speed is difficult, but once you are in the middle of the water it is smooth as silk. The transition from ground to cruising altitude is similar. The initial climb will probably be turbulent but the aircraft is built to sustain more than nature can throw at it. Once at altitude the wind speed will almost certainly be much more than 100 knots - but just like the fast flowing  river, if you close your eyes it will feel nice and smooth. 

You should only cancel your flights if you want to throw away your money. Otherwise just try to relax, and understand that many airline delays that you hear of are caused by a pilot choosing safety over schedule.