Seems high to me. House prices are $130-$150 per square foot right now in many neighborhoods - about the same price as you were quoted for merely wood. I think you need another bid. If I were you, I'd ask for an itemized quote, showing materials costs separate from labor. That will show you the breakout.
Framing a basement of 800 sq ft should take a competent professional just a few days with a small crew - maybe 24-48 man hours, unless you have some really odd setup. Shucks, I framed a 1200 sq foot single floor by myself in a few days, including sheathing the outside; I only had help standing up the walls. I can see paying the foreman or General Contractor $100/hour, but his laborers shouldn't be more than $25/hour even with benefits and insurance ($10 if they're undocumented....). So I would guess labor would top out at a couple thousand.
As to materials, studs at Lowes right now are $2.50 each; top and bottom plate is $0.40/foot. Figure on 1 stud every 16 inches of wall, plus top and bottom plates, plus some kind of structure for the ceiling. Add 50% excess for corner posts and the like, and for extra structure like headers. Assuming 24x36 feet, in four rooms so one set of crossing walls, that's about 180 feet of wall, for about 135 studs. Add 50% extra for corners and waste gives you about 200 studs, for $500. Top and bottom plate is another 540 feet, for another $220. You can probably make it cheaper by furring out the walls rather than full studs. I doubt framing for a full ceiling is required, but you'd have to add a suitable amount for some kind of ceiling framing if you plan to drywall it and you can't simply apply the drywall to the bottom of the first floor joists. So add some suitable amount for boxing around HVAC and plumbing and wiring as required. Add more if you have a lot of walls in mind, and it's not just a big open space.
So even assuming a lot of lumber I haven't thought of, that's still around $1000 for lumber, or maybe $1500 if you need to frame the ceiling. (And you can do far better at 84 Lumber with a contractor's account.)
So I guess I'd be looking for a bid around $4000 tops.
No, I'm not for hire.