The striker scored both goals on a rollercoaster night at St Jakob Park.
We travelled to Switzerland level at 2-2 but soon forged ahead with Clint's opener. That lead lasted four minutes and FC Basel went 2-1 up on the night, 4-3 ahead on aggregate early in the second half.
We switched to three at the back in search of an equaliser and Clint was there again to level the tie at 4-4 seven minutes from time.
Jan Vertonghen was then sent off but the 10 men defended stoutly to take the tie to a penalty shoot-out – won 4-1 by Basel.
With our Premier League opponents Chelsea in the FA Cup semi-finals this weekend, we now have a break until our next fixture against Manchester City at the Lane in Sunday week, April 21.
That kick starts the run in of six matches as we fight for a place in the top four.
"The players are upset but like in life, when you get knocked down, you have to focus on the next thing," said Clint.
"For us, that's the Premier League, finishing in the top four and making sure we get into the Champions League next season.
"We're all disappointed now but we have to regroup and focus on the Premier League."
As for the events in Basel, Clint added: "We showed a lot of character to get the second goal and playing a man down in extra time was really difficult, but we pushed ourselves to get to the penalty shoot-out. Once it got to penalties, you know anything can happen and we were just unlucky."