A browser recommendation is usually a Do Not Enter sign. Unless you have an extremely popular service with unique features, it is extremely unlikely that people will change browsers for you. Therefore they are highly likely to simply leave.
This does not mean you have to write code that works perfectly for every browser. In a support page somewhere you can have a list of tested browsers. Only people seriously interested in your service and experiencing problems will end up there. Those are more likely to listen to you.
The best thing to do is to look at your stats and see which browsers are most used to access your service. You should support at least 80% of visitors, I recommend 90% even but that depends on the difficulties you may have to support certain features on less common browsers (some of them may be fairly new even).