Basement – Unfinished basement is open to vented crawlspace, what should we do to finish

First, handle drainage. If you want to install a sump pump, perimeter drain, or water proof the walls, now is the time to do that. You should also install any plumbing drain lines at this point.

Next, framing. Concrete transmits moisture, so use pressure treated, and a styrofoam underlay that would normally go under the sill plate, to keep the walls dry. With pressure treated wood, you need galvanized nails to avoid a chemical reaction that would eat a normal nail. Make sure the framing is designed for the drywall (16" OC studs, nailing edge in the corners and around the ceiling perimeter). And make sure the walls are positioned/sized for any plumbing or other utility lines. For exterior walls, the thicker the wall (2x6 instead of 2x4) the more insulation you can include.

At this point, utilities go in, including plumbing, electrical, hvac, and communications. I'm a strong believer in conduit for running communications lines (like ethernet, catv, and phone) so that you can upgrade those without opening the walls or running wires all over the floor. Note that with more outlets, you will need more circuits, and copper isn't cheap.

Insulation is done after all the utilities are finished. If you have water proofed the walls, then I don't have much against the standard fiberglass insulation. Actually, I like fiberglass because it's easy to work with as a DIYer, relatively cheap, and replacing a piece or temporarily moving it (e.g. in the attic) is simple. However, with moisture concerns, or if you live in a northern climate with harsh winters, then I'd favor the spray foam. The spray foam has the downside of being difficult (impossible?) to remove, but it functions as a vapor barrier and leaves no cracks that air can get around.

After this, you have drywall, paint, doors and trim, and touch-up paint. I like to paint the walls and trim separately so there's less work on edging.

As for the flooring, you would usually do this before or after the trim. With hardwood, you would often do the flooring first, and then the trim goes up tight against it. With carpeting, you would install your trim 1/2" above the floor and the carpet installers would install the carpet right under that.

It is a good idea to seal any apertures between the crawlspace and the main house. If you have to rely on caulk and spray foam you're probably in a situation where the house has some weird structural problems - that is, you should not have giant gaps between the crawlspace and the floor above in any normal construction regimen.

It is not a good idea to seal off the crawlspace, ventilation is not only a good idea but it's required by code in locations where crawlspaces are an accepted norm (note that in places where winters are exceptionally cold the norm is finished and unfinished basements, not crawlspaces). The fact that you have had mold in the crawlspace is actually an indication that you might need MORE ventilation not less.

A proper insulation/vapor barrier installation at the level of the floor joists should solve any draft problems. Also, looking at your photos I would recommend that you insulate your ductwork/boots/wyes per local code.

So to reiterate, please do insulate and install a vapor barrier! Do not seal off the ventilation for the crawlspace - indeed look into your local code to see if you have enough ventilation.

As a side note - any issues of moisture (and therefore mold) in the crawlspace need to be addressed. Possible sources of moisture include but are not limited to: leaky plumbing supply or waste lines, inadequate drainage system at the perimeter of the house (undersized gutters/downspouts, poor grading, no storm drain, etc.), and a possible high water table.

Good luck staying warm this winter!