Can anyone please recommend the best way. Approximate taxi fare, if you know? Thank you!

I used to live there and if I remember correctly, it's a bit of a drive for a taxi. It's easy to find down the main highway. Is it 95? I can't remember, but the signs are clear along the way. My guess is a taxi would be expensive.

Maggie-
The Light Rail line literally stops at the airport (Pier E). It will take you directly to the Inner Harbor area.

I agree with Larry. BWI is at the END of that lightrail branch, so it will be almost empty when your board. Buy a ticket from the machine just outside and to the right of the pier E exit doors on the north side of the terminal. Fare is $1.35. Get off at the Convention Center stop (Pratt St), and get a taxi from there to your destination if you need to.

You see airline crews all the time taking lightrail to/from the airport.

Taxi should cost about $25. Some hotels offer direct shuttle service, call your hotel to find out availablity. Also in the baggage claim area, you can pick up a hotel shuttle that stops at a variety of hotels around the harbor for about $15. Go to BWI website for info, I am sure there is information on this service.