Even if you're running a serious game. Even if you're in a serious part of the game. Don't forget that it is ok to be a little silly, and that silliness might even be something that just happens. Some players - like myself - will occasionally make jokes to lighten the mood when things seem to be getting too tense or dour. It doesn't mean they're not enjoying things - if anything it means you're having the desired effect.
Consider also that RPGs have an inherent silliness to them. This is a game of make believe and pretend. Where else do you have a 40 year old office manager pretending to be a Dwarven Wizard complete with silly accent and trying to act out - with no official acting lessons - the hardship of getting past said dwarf's gruff emotional exterior?
Finally consider that human interaction in the real world is silly and awkward at times. Hollywood, comics, books, and other works of fiction like to show us that everything can be tense, dramatic, and perfect. Fight scenes are choreographed. Sex is sensual and/or physically adept and smooth. Heart to heart conversations stay on topic with no one talking around what they're trying to say. But that isn't how these play out in real life. In real life these exchanges can be awkward and silly at times, and you have to embrace it.
And part of that when it comes to fiction is scene setting. Unless you're a god of sound design for your RPG and piping your sound choices to the game with custom precision for the scenes, odds are some of your tense, tearfilled, emotional moments at the table will become silly. Don't believe me? Check this behind the scenes video from the development of Dragon Age: Origins, where scenes that were flat or silly suddenly became dramatic and tense when the right music was added. Then they added music and suddenly the same scene hit emotional and heartfelt. This you can do in your games, but even having playlists for emotions can fail when a song hits an offbeat for what is going on specifically in your scene.
Ultimately there is no reason to fear a bit of silliness. You can talk about things and emphasize. And above all, if a scene needs to be taken seriously you can always ask people to let the scene play as this scene is important. But mostly as long as things stay on topic and move forward, let people be silly. Let the game be fun. Maybe it won't be the epic fantasy you envisioned...but that's not what games are. If anything games are closer to the "here is what actually happened" and then the bards go back and revise to make it the epic narrative people buy at the airport and read.
Have fun. Keep things moving. And don't fret if games become a bit silly now and then.