Larger logs, when split open, will probably have some dry wood towards the center. This can be shaved off to help create tinder (and possibly kindling, depending upon the thickness of the log and how long it's been wet for).
 

Certain types of pine and sappy softwoods occasionally develop centers of sapwood that some people call "fatwood" - it usually looks waxy or oily, and will burn quickly even when wet.  

 The most important thing is making sure that you have enough DRY tinder to both dry out and light your kindling (cut your kindling thinner than usual to facilitate drying it out). If you have dry tinder with you, great. If not, look for fatwood (see above), birch bark (from a fallen tree if possible, but if it's really a survival situation, I'm not going to fault you...), and any dry stuff around. Twigs lower on a tree and closer to the trunk will probably be driest. Evergreens (particularly firs) are good to find dry wood on, even after a few days of rain.

If something is already dry, by all means, KEEP IT THAT WAY! Once your fire gets to rolling, you shouldn't have any problem. Dry wood out by laying it next to the fire, and use smaller pieces of fuel wood to keep it going. They'll dry out better and hopefully prevent
smoldering. 

I recommend that they keep a fire starter in their survival kits. It's amazing how much help a wax candle stub can be in damp weather. Having some newspaper or solid lighter (meta tablets, cardboard impregnated with paraffin, etc. helps a lot. ). Remember the basics, tinder is what starts a fire, kindling starts the fuel. With wet wood you need lots of tinder and kindling, Split the wood, or break it or dig into it. Get at least Two LARGE handfuls of tiny splinters and shavings, four is better. Make sure you have
LOTS of kindling about pencil thickness and some that gets gradually larger to about 3/4 inch diameter. Build your lay carefully, the younger scouts today, doing most of their camping with stoves, have the hardest time with this part.