What sort of things can I keep at basic?

Bring your toiletries, if it's still the same, you'll have a "personal drawer" for those things. Anything specifically forbidden they'll throw away or lock in your personal bag, in a secure room.
Depending on your length of AIT it'll either be the same as Basic or they'll relax the rules. Your best bet is to follow the packing list and pack light. They'll give you everything you need.

You should receive a recruit trainee packet that spells all that out. It will list EXACTLY what you are to bring and if you FAIL to bring that then upon your first trip to the PX you will buy said items out of your first allotment of pay. Except where noted such a list is 1 (one) each and any extras will be stowed with your personal effects which you will not have access to until graduation or move onto the AIT portion of training.

Baby wipes may be on the list that is allowed. Check with your recruiter. They are most likely considered a toiletry/personal care item and allowed. I know we had them back in the day for rucking and field problem, along with baby powder.

Assuming you are OSUT, one stop unit training, then your personal effects that were stripped of you in basic will be returned upon going from basic phase to AIT phase. Once you hit AIT then you will have SOME extra liberties and freedoms such as free time to go off post etc.

Check your recruiter or futuresoldiers.com to get a list of what you can to basic. Make sure you don't bring any aerosol products or cleaning stuff in a different container like lotion poured into a little bottle.