It is the second tie during a three-game unbeaten stretch for the Rapids.
Javier Lopez scored the goal for the Earthquakes, who saw their winless streak (0-6-3) stretch to nine games.
But the result did end San Jose's five-game losing streak in Colorado. However, the Earthquakes haven't won in their last 10 visits dating to a triumph in June of 2013.
San Jose had a chance to win the match in the sixth and final minute of stoppage time but Chris Wondolowski's right-footed direct free kick sailed over the top of the net and the referees immediately signaled that the contest was over.
The Rapids have scored a goal in 31 straight home matches, tying Atlanta United for the sixth-longest such streak in MLS history.
Colorado was missing forward Jonathan Lewis and defenders Kellyn Acosta and Sam Vines as the trio is playing for the U.S. in the Gold Cup. San Jose midfielder Jackson Yueill also was away playing for the U.S.
The Rapids had a 14-7 edge in shots on goal but placed just two on target. The Earthquakes also put two on target.
The Rapids knotted the score in the 53rd minute as Nicolas Benezet set up Bassett, who delivered a left-footer into the left corner to beat Earthquakes goalkeeper James Marcinkowski.
Bassett had a chance at a second goal in the 78th minute but sent a right-footed shot over the goal.
Colorado's Auston Trusty sent a left-footed shot well over the net in the first minute of second-half stoppage time.
Marcinkowski and Yarbrough each made just one save.