That's NOT GOOD. (& it's more important than your other power-steering business.)

You can only drain about 1/3 of the fluid, since the torque converter doesn't drain. So drain/fill, then drive a few miles & do it again. Buy about 10 quarts & do it again & again until it stays clean. When you open the first new bottle you'll know what color it's supposed to be.

Check your manual about fluid level. Honda's a little weird. Most others you check with the engine warmed up, idling. My '98 Accord wants it warmed up & turned off. I think most Honda products are like that.

edit...
It just occurred to me. How many miles on that 2003? What's the warranty situation? Even if you change your own fluid, I'd certainly get on record at the dealer about the fluid turning brown.