The Swiss team progressed at Liverpool’s expense, holding out for the point they required to take second place behind Real Madrid in Group B.
Fabian Frei gave Basel a deserved first-half lead and, by the time skipper Gerrard netted a superb 81st-minute free-kick, Lazar Markovic's red card had reduced Liverpool's prospects on a must-win evening at Anfield.
"We haven't gone out of the competition because of our performance tonight," Gerrard told ITV.
"We've gone out because away in Basel we weren't good enough and in our last game we conceded a stupid goal towards injury time.
"You always qualify over the six games and unfortunately we haven’t been good enough."
A lengthy injury list has contributed to a lacklustre campaign for Liverpool and Gerrard feels a shortfall in attack - with Daniel Sturridge and Mario Balotelli currently sidelined - is hampering their efforts to build on last term's runners-up spot in the Premier League.
"We need to get our main striker [Sturridge] back, who is capable of getting the goals to get us into the top four," he said.
"We need to try to get Mario Balotelli back firing as well - if we can get goals in the team it gives us a better chance.
"At the moment it’s a bit of a rollercoaster, it's tough, but take nothing away from the lads who are here, slogging it out every single week.
"Rickie Lambert ran himself into the ground over the last five games and unfortunately this game has come as too many in a short space of time because we’re short on numbers.
"But, of course, this club always tries to strengthen, it always tries to improve. We dust ourselves down and go again."
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