http://en.wikipedia.org/wiki/Belgium#Languages
says that 59% of the population speak Dutch and 40% speak French. However the vast majority of people in Flanders natively speak Dutch and similarly the vast majority of people in Wallonia natively speak French, apart from the small German speaking area in the far east around Eupen. So if you are absolutely sure you are in a Dutch or French speaking area then the native language could be useful, but certainly not essential for a tourist. In most of the country, signs will only be in the one native language.
Brussels is the major exception as it is officially bilingual with French and Dutch having equal status, although French speakers are in the majority. Signs in Brussels are in both languages. Given that random people you meet could speak either language, speaking in English is actually preferable to avoid speaking French to a Dutch speaker or vice versa.
There are also a few places near the language border where 'language facilities' exist for either minority Dutch/French speakers, and signs may be bilingual in these areas.
How do Belgians start a conversation with someone where there is a good chance that someone may be a native speaker of either Dutch or French? Do most simply speak their native language? I try to ask a short question in both languages and listen for the person's reply before continuing but would be interested to know what Belgians for example on a train between Bruxelles-Midi/Brussel-Zuid and Antwerpen-Centraal would do.