To me, the main evidence the characters can see phaser beams as visible light is the general way they act around hand weapons...
Hand weapons operate in atmosphere. It's plausible that an intense particle beam could excite or ionize the air it passes through sufficiently to make it glow, the same way a lightning bolt does. I'm talking about energy beams in vacuum, which is a very different physical scenario.
The only way a beam can be visible from the side is if light is being emitted toward the sides. Since a beam, by definition, consists of energy or particles moving in a single direction, beams by themselves are normally invisible from the side. The only plausible ways to make a beam visible are a) to pass it through a medium dense enough to scatter its light to the side, as when filmmakers use smoke or mist to make lasers and flashlight beams visible, or b) to pass it through a medium whose particles are excited or heated to the point of luminescence. This is why visible beams in vacuum specifically are so implausible.
(Although excited or ionized air in the path of a particle beam wouldn't look anything like the energy beams of TV and movies. The glowing air would be a heated fluid, expanding and rising, so if the beam were sustained for any length of time, the visible effect would have a liquid, flamelike quality to it, like a sustained electric arc. That's something I would personally love to see depicted realistically, because it would be a thousand times cooler than the cliched energy beams created by FX animators.)