It wasn't ever stated on TV. (thousands upon thousands of ships, maybe millions) Considering that every world in their territory served them and we only saw a couple of different craft.
In one episode of DS9 it was mentioned that in the second year of the Dominion War the combined forces of the Dominion, Cardassians and Breen in the alpha quadrant numbered around 30.000 ships. Presumaby the majority of these were Dominion ships. It is unknown how many more the Dominion had in the gamma quadrant, although we know they were able to send at least 2800 ships without fearing for the safety of the Dominion in the gamma quadrant. Since the Dominion was engaged in a major war and could build ships very quickly it remains unknown how many ships they would normally have in peace time. However, given that most Dominion warships were small attackships (bugships) that were easily destroyed by the, on average, larger and more powerful Federation, Klingon and Romulan ships and judging from the known balance of power (Dominion > Federation > Klingon = Romulan > Cardassian =? Breen) I'd say the Dominion would need at least 20.000 ships in peace time to be a significant threat to the alpha quadrant (I believe the Federation has about 8000 starships).