The Rapids (5-4-4, 19 points) have won two straight and three of four.
Bassett (35th minute) and Jonathan Lewis (50th) made it 2-0 before an own goal by San Jose's Tommy Thompson extended the lead to 3-0. Nicolas Mezquida scored in the 79th and Andre Shinyashiki tallied five minutes later to add to the Earthquakes' misery.
San Jose (2-6-5, 11 points) has conceded 11 goals in the past two matches and has three losses by at least five goals over the last two weeks.
San Jose, 0-5-3 in the past eight games, was coming off a 6-1 loss to the Portland Timbers on Saturday. Earlier this month they lost 5-1 to Los Angeles FC and 7-1 to the Seattle Sounders.
The Earthquakes have the worst goal differential in MLS at minus-21 (17-38).
"We have a group that has experienced these times before. For a long time we've been changing that mentality. Today we went back to the old mentality. It was very difficult," San Jose manager Matias Almeyda said via a translator.
William Yarbrough made four saves for the shutout.
On the opening goal, Bassett timed his run to await a free kick by Jack Price, who assisted on two goals. Bassett was alone to snap a header for the 19-year-old midfielder's second goal in as many games and fourth of the season in nine games.
The goal came six minutes after Colorado's first goal was disallowed following a video review that showed Price was out of bounds when touched the ball.
The Earthquakes' defensive lapses were evident when Lewis tallied his team-best fifth goal. A passing sequence allowed Acosta to slip a through ball to Lewis scooting down the middle and he finished the play.
Colorado stays home to play Sporting Kansas City on Sunday. San Jose plays at Los Angeles FC on Sunday.