Tbh, OP, given the rarity of 386 boards, and the fact that most of what is left is 386SX, I'd sooner focus on what is available to you within acceptable distance and budget, and then choose from the options you have available.
If you are lucky enough to have multiple options, a lot depends on what you want to do with the system exactly. Do you prefer to constantly mess around with hardware, or do you prefer to build once, close the case and then only get involved with the software?
If you like messing around with hardware - like most of the people responding here, me included - you should probably look to the most versatile board you can find. I'd suggest a 386/486 board, which gets you very high performance, cache, VLB and the option to run both classic 386 CPUs and 486 but more interestingly most of the DLC souped-up 386 CPUs. Look for something like the Auva CAM 33-P9, or more generically anything with the OPTi-495SX chipset which was invariably used for those boards.
However given your topicstart I suspect you're not that interested in constant tweaking. In that case, a 'classic' 386 board with regular 387 co-pro socket is probably best and will definitely look 'cleaner' than one of those late jack-of-all-trades. The first Asus boards (ISA-386xxx) are generally considered excellent, but again, chance of finding something specific is minimal, so just look at what you can get and choose there.