Would probably make your life easier to remove the baseboard first. Reinstall after.

Half my basement is 3/8" rubber. Workout/storage room and my toy room. Without the baseboards removed, I am not sure how you would put it down effectively. Simple baseboard is dirt cheap and easy to install. 12 or 16 foot contractor packs are $60-$80 primed. Typically, the only floor you install with the baseboards on would be carpet.

Your last option, and I say this weakly because for certain things like you are trying to do it is not the correct method of install in my opinion, you could install quarter round or shoe molding along the baseboard and simply run the rubber up to the baseboard.

With that said, get double sided carpet tape that is rated for concrete to stick it down well. Not cheap, but you will thank me later. Standard/cheap carpet tape will likely not hold up well on concrete.

Really? I have a new, custom home. 7" baseboards. With quarter round.

Quarter round or shoe mold is the correct way to finish it. I worked for a flooring contractor for awhile.
You'll also want adhesive for your flooring. The flooring should have a label indicating what to use.

Should have phrased this differently. Just running the rubber up to the base molding and simply installing shoe or quarter round is not the best method, IMO. I think it looks poor and could leave you with potential gaps at the wall if not all the way to the base molding. In this case, I don't see a reason not to pull the base off the wall then reinstall. The final product will look better. Whether or not you want to use quarter or shoe after that is personal preference. As a reference, I think till looks like crap with out shoe or quarter on the base molding. However, till will typically be ran underneath the base.

Thanks for the tips! The baseboard was installed 5/8” off the floor using the scrap edges so we could put the flooring under it. It actually worked way better than I thought. I removed the scraps and caulked and painted it in place over the cement floor then slid the tiles under it.

The job was a little weird since we did a lot of it ourselves but due mostly to my husband’s work schedule I hired someone to install the French door and trim. The guy who did the door and trim suggested doing it this way (I think he just wanted to knock everything out in a day and not have to come back).

We are contemplating redoing the flooring in the whole basement at some point - so wanted to be able to remove fairy easily if we decided to do vinyl plank through the whole level. So we did not glue down. Will we have a problem with it just sitting on the cement?