Totally depends on how you count. Of all the videogame publishers Nintendo has been around the longest, but of course for half of the company's life it was making playing cards. Sega have been around since the 50s (I think) but again spent their formative years distributing mechanical amusements and not videogames. Atari have been around since 1972 and Activision since 1980 but of course neither are really quite the same company they were when they were formed.
The best I can figure it, if you count the modern Atari as being a continuation of the company that installed Pong in Andy Capp's Tavern then clearly they're the longest running. If not, I reckon it's Taito who first dipped their toe in the videogame market in 1973. They're swiftly followed by Nintendo who's first step into gaming was to distribute the Japanese version of the Magnavox Odyssey* which they launched in 1974. I believe Sega's first videogame was launched in 1975 which probably gives them the bronze.
*Which technically doesn't make them a 'software' publisher, I guess. But the Odyssey shipped with a variety of cards which played different games. The games were all pre-loaded on the console, the card just told it which to play - but at a push I guess you could call the cards 'software'. At any rate, since we're including arcade games (which are also solid-state machines which play only the software they're designed to play, at least in the pre-JAMMA/Neo Geo era) in the reckoning here I don't see why we can't include the Odyssey.
Owned: NES, SNES, N64, Gamecube, Wii, Game Boy, Game Boy Color, Game Boy Advance SP, DS Lite, 3DS, Master System II, Mega Drive II, Saturn, Dreamcast