In order for this not to become a problem in the long run, there would have to be goods/services that could only be (or at the very least are much better to be) purchased in bitcoins, forcing people to exchange fiat currency for bitcoins and closing the trade deficit. Maybe this isn't a problem at all, once the bitcoin community becomes big enough a large enough number of people will prefer to store their wealth that way that when they are paid in FRN they will exchange that to bitcoin?
There is no easy solution for that trade deficit and I'm quite sure that in current times people still prefer fiat over Bitcoin for purchasing of goods at local stores or in online stores. There aren't any good online markets where users can get unique items only using Bitcoin and that's not going to change anytime soon. Bitcoin is getting accepted by many new people and it will continue to get accepted by even more people overtime, and there are chances that in future some websites or a local businesses might accept Bitcoin only payments for some specific items.
But, still I don't think that any reputed businesses would take a step like that because disqualifying fiat payments could reduce the number of customers for those businesses and that's why any business or site would probably take a route where they will have both Bitcoin and fiat as way of payment. They can instead have a strategy in which users that pay via Bitcoin get some extra perks, bonuses, and discounts, and a strategy like that might increase the trades with Bitcoin but still that approach will take a lot of time.