For germinating seeds, I usually put a thin layer of clingwrap over the top of the pot and secure it to the pot. This traps the moisture.

Yeah, I usually do that too, but these are in big planters outside (same pots as my tomatoes and cucumbers), but maybe I can do the same somehow.

My carrots are planted in a plastic storage bin. When I first sowed them, I wrapped plastic sheeting around the entire bin, longwise, tucking the edges under the bin. This left a little space on the long edges for air and some rainwater to get in. Oddly enough the initial germinations were closer to the spaces rather than the protected middle.

I've been growing carrots in containers for the last 6 years. Never used plastic on top or worried about anything but keeping the soil damp (which has been very easy this year, since it's rained so often. lol)

My carrots sprouted! Yay!
I actually took the covering off after a few days, because we've actually gotten a decent amount of rain and probably it's better if they can breathe.
Now to wait and see if they turn into real carrots. I might start a few on their own in a tall container and see how that goes as well.
Does anyone know if they'll grow ok inside? (in a very very sunny spot?)
Peg