How bad are small core shots on the ski base?

It depends. Ive skied on skis with many core shots and it didn't affect me. If water starts getting to the core than it could be a problem

lilmayoGot a core shot in my ski this weekend, right next to the edge without the edge being broken, will this cause any problems for the skis in the future?

Well if it actually is past the base and into the core, then seal that shit up after drying sufficiently. Don't want water in your core

^yeah does it actually go through to the wood? Deep gouges are no biggie if it doesn't actually go through. If it is right next to the edge there is a chance that you will hook your edge on something though. Super easy to drip some ptex in there though just be careful because flaming drips of ptex are actually hot enough to catch bath towels on fire. lol

lilmayoGot a core shot in my ski this weekend, right next to the edge without the edge being broken, will this cause any problems for the skis in the future?

Fix it or get it fixed! Here you go! https://youtu.be/suk5BFA7ETc?t=30m50s