Can you clarify what you mean by the "bottom of the seat tube"? Do you mean the end of the tube where the seatpost goes in or do you mean the bottom bracket shell (where the seat tube joins with the town tube and chain stays? If it's the latter, this is the area of the frame where rust is the biggest concern. There are various rust removal products on the market, some really toxic and some not so much. I've read about good results with Evapo-rust. My understanding is that you need to be sure to dry the bike immediately, rather than let it air dry, or rust will quickly develop again. From there, apply rust inhibitor inside the frame and you should be, hopefully, good to go.

But I'm no expert on this having never done it myself. My DH knows more about it than me; I'll ask what he recommends.