Go to Google Maps and enter "Parking Smithsonian" into the search field and they'll all be displayed on the map. Reasonably priced? I think they all are, but I'm used to city parking prices. When you factor in the cost of the museums (nothing) it's more than reasonable. lol
Have a great time!
Reasonable, probably not. Near the Smithsonian? Which one? There are 13 museums and galleries both on & off the Mall in the Smithsonian Institution (that's only in the DC metro area, there are more in the surrounding region), so regardless of which one you park nearest, you still will have potentially several miles of walking between/within.
You might consider parking on the outskirts near a Metro station and taking that in. Probably a lot less stressful as well, compared to contending with the DC traffic at the wrong hour(s) of the day.
Here's a place to start to get an overview: Maps and Directions | Visit | Smithsonian
I would recommend to check Washington DC Parking Search Engine - Find Daily & Monthly Parking for the parking fees.
Also, in DC for $7/day you can run a bike: Capital Bikeshare. They have a lot of stations, but with typical problems like no bikes or full stations (so you can't return it). Luckily it doesn't happen very often.
I rarely drive in DC, even all the years I lived there.
One of the best things about DC is being able to walk or use the mass transit.
I would look at a satellite parking lot at a metro station, and leave the car there, and use the metro to get to the smithsonian. Its alot easier to enjoy if you are always thinking about the car.