You Won't Regret paying for the transfers
Money is money, but in four years of working with travelers to Russia and the former Soviet Union, everyone who took public transportation after the long flight has regretted the decision (not due to safety but because of comfort and convenience). This is even more true for people with no real knowledge of the Russian language.
As other posters have said, either pre-arrange the taxi with the hotel (which is expensive) or book the taxi at the terminal when you arrive (usually this is reliable, although I have heard a few problems with this as well, mostly due to miscommunication rather than ethics). There are also agents in the US who can do this for clients, usually mid-priced but very reliable.
There are 4-6 Marriotts in Moscow (depending if you count the Ritz-Carlton and Renaissance as Marriotts) so make certain you give the driver the address of the Marriott Tverskaya (the Marriott Grand is also on Tverskaya, closer to Red Square).
Marc David Miller