Michael Claassens and Jack Cuthbert crossed late after their forwards had earlier earned a penalty try, but they rarely looked like having enough to deny the French outfit who led from the 16th minute.
"It was very disappointing. It was a big game and a big occasion and it didn't go the way we would have liked," said forwards coach Booth.
"From a physicality point of view, we were found wanting, especially at the breakdown. They scored three tries from turnovers.
"The bottom line is that we weren't good enough on the day and Stade deserved to go through."
Bath battled back into contention after trailing 20-3 at the break but Stade were always just too far ahead, with Booth pinpointing his team's failure to hang on possession at key moments as a major factor in their exit.
"I thought we were credible and fought hard to get back into the contest but we couldn't really bridge the gap that you need to make it that little bit jittery," added Booth.
"One of the number one basic rules of attack is that if you look after the ball it'll look after you, and we didn't do that well enough in the first half and it made it very difficult for us.
"All they do is attack, so we knew going into the game that if we didn't control the football in attack and didn't defend well, things were going to be difficult for us.
"We didn't get cut too much off first phase, it was all off turnovers and individual brilliance - that's what the top sides to do you.
"If you've got world-class finishers and you give them time and space they are going to hurt you, and that's exactly what they did."
Bath's defeat completed a hat-trick of quarter-finals losses for English clubs after Gloucester and London Wasps also went out to Biarritz Olympique and Leinster respectively.