Hi Sid,
Thought I'd try and help you out with this one. Without knowing what dates and class you are wishing to fly, I thought I'd quickly look at the price difference and duration of trip as a comparison. I chose the outbound date of February 15th and the return date of the February 22nd all trips in economy.
Emirates:
EK232 IAD-DXB
EK506 DXB-BOM
EK509 BOM-DXB
EK233 DXB-IAD
Trip duration for IAD-BOM is 16h 15m.
Trip duration for BOM-IAD is 20h 30m.
Total trip cost is $1,154.12 including taxes.
British Airways:
BA66 PHL-LHR
BA139 LHR-BOM
BA138 BOM-LHR
BA67 LHR-PHL
Trip duration for PHL-BOM is 19h 50m.
Trip duration for BOM-PHL is 23h 00m.
Total trip cost is $1,248.32 including taxes.
So from the prices and duration its evident that with Emirates you'd save just under $100, but at the same time save your self 6h 05m of flying during the whole trip. So even if your driving took 2 hours each way, you can always know your still saving time by flying Emirates
Service wise, I'll have to be perfectly honest and tell you I've never flown British Airways so I can't benchmark their service against Emirates'. However I've flown Emirates in Y,J,and F many times during the last few years and can assure you that Emirates' service is significantly better than what you'd expect on any US airline. Your Washington to Dubai and Dubai to Washington sectors will be on Emirates' Boeing 777-200LR which has all the latest and greatest entertainment system, and I think its important that on these long flights you keep your self entertained otherwise you'll just end up plain bored. I'd recommend paying the extra for business or first since you'll want the sleep!
The Dubai to Mumbai and Mumbai to Dubai sectors will be aboard Emirates' A330-200 aircraft which doesn't have the best product, however you'll still experience great service.
If your American you won't need any visas for transiting through Dubai. Also the connection times between flights above are all adequate.
Hope this helps you decide, if you have any further questions reply here or PM me!