Fabian Frei gave the Swiss visitors a deserved lead as Liverpool produced a disjointed first-half showing in the vital Group B meeting at Anfield.
The introduction of Lazar Markovic at the break brought an improvement from the hosts in a game they needed to win but the Serbian winger was sent off with an hour played.
Replays suggested Markovic's raised hand made minimal contact with Basel defender Behrang Safari but Kuipers had little hesitation in producing a straight red card.
Steven Gerrard scored a superb free-kick to set up a grandstand finish, during which Liverpool pressed hard for a winner that they were unable to force.
Rodgers must now prepare for a drop down to the UEFA Europa League in the new year and he had little time for Kuipers' display after the match.
He told ITV: "I thought it was an awful decision. I though the referee was disappointing on the evening to say the least.
"Markovic came on and was bright. He made a difference to the team, he was dynamic.
"He's just taken a touch away, he's obviously got his arm up. If that's a sending off you're going to get two to three sendings off every single game.
"It was a really, really poor decision. I'm not even sure if the referee has seen it.
"I think he's seen the player hold his face and lie down but there's no way that was a sending off."
Nevertheless, Rodgers accepted the overall blame for elimination lies with his own team, who went five games without victory on their return to the competition having scraped a 2-1 victory over Ludogrets in the group opener.
A sub-standard start to the Premier League season means Liverpool face a battle to return to European football's top table but Rodgers is confident it is a fight they can win, with a spot in next year's competition on offer if they can win the Europa League.
He added: "As I said before the game, the players deserved the opportunity to be in here and we have to fight, going forward, to be in it next season.
"Obviously our performance level needs to improve over the course of the season but we're not too far off the top four.
"If we do that we’ll be in the Champions League and we'll be better in it next season."
