I want to put dimensions into a section in assembly draft, but a manually placed hatch keeps covering them.
A hard workaround is drawing rectangles by hand around every dimension, but for obvious reasons I'd like to avoid that.
Is there a setting I'm missing or a more automated workaround?
Yes as bshand mentioned this is the best method to get that.
Set the "Fill Text Background" as default then all the dims will be seen better.
Not only if hatch is behind but also when other lines, dim lines etc. are there.
There is one additional hint I would like to mention.
Although it is set to be filled with color it could be that Your dim text will be overlayed by the hatch.
Then You should put the dim on top of the stack using the command.