I've lived in Virginia. I now live in Washington, where the minimum wage is enormous. Funny, so are the prices. I can't afford anything in Washington. I wish I could move back to VA, where I had a better standard of living: my lower salary stretched much further. I had a gorgeous three bedroom apartment for 700 in Virginia; I have a shitty one bedroom apartment for 1000 here. Food was cheap in Virginia; I skip meals here in Washington to save money.
Are the two linked though? If you're in Washington DC, there's a limited amount of land, which should increase costs across the board.
Contrast with Washington State, which has a similar minimum wage. Yet if I look at rural Washington State Craigslist (Moses Lake, specifically - I presume that's rural enough but I'm not from the West Coast) - I find 2 & 3 bedrooms for under $700.
In my state, if I took what I paid for my small, modest house and went out into a rural area, I could get 4x the square footage for half the price! No minimum wage difference, but there is a difference in how populated each area is.
There is probably some link between minimum wage and higher prices - in both directions. A higher staff salary will bump up the prices slightly, but in an area where the COL is high, wages may be high enough already that there isn't a strong enough opposition to raising the minimum wage. The same holds true for areas with a low COL - wages may be low enough that businesses fight against raising the minimum wage.