It was Mourinho's first European game in charge of the club since his return this summer and Chelsea looked on course for victory when Oscar opened the scoring at the end of a tight first half.
But the Blues squandered a number of chances to extend their lead and were stunned by late goals from Mohamed Salah and Marco Streller.
The Blues were also beaten by Everton in the Premier League at the weekend, but Mourinho insisted after Wednesday's loss there was no need to panic.
"I am the manager, I am responsible for everything," he told Sky Sports.
"Especially after a bad result. In the good moments I want everyone to shine, in the bad moments I want everyone to keep calm.
"Tomorrow I will not wake up with a smile on my face but at seven o'clock I will work hard to get three points against Fulham.
"In the next match, we go to Bucharest (in the Champions League), we have to get the points to put us in the Champions League. We don't want to play in the Europa League."
Mourinho was impressed by Basel's performance but was left to bemoan his side's failure to make the most of their chances.
But while few had expected Chelsea to lose, Mourinho was not too perplexed.
"I not in shock because sometimes we win, sometimes we lose," he said.
"I am not happy of course with the result because instead of taking one step in the direction of qualification, we did one step back, but we have to assume the responsibility that we have five matches to play in the group phase."
Asked where it had gone wrong, Mourinho added: "I think they played well. I think in the first half they defended well because we had a lot of the ball but could not create many chances.
"Oscar scored at a good time for us and then I think they scored their equaliser at our best moment of the game, after we had a couple of good opportunities, and after that we were punished defensively."
Mourinho has placed emphasis on youth, even though his team has enormous experience, including Samuel Eto'o, who was ineffective.
Mourinho stressed his happiness with his strike options, after leaving Fernando Torres out of the squad and bringing Demba Ba off the bench, but admitted Cameroon forward Eto'o may be lacking the sharpness displayed for Barcelona and Inter Milan after a spell in Russia with Anzhi Makhachkala, where he commanded a vast salary.
Mourinho said: "The sharpness, that click to score, we have to wait. But he's a great player and he will score goals."