Investors were rattled by gains for anti-austerity parties in Spanish local elections over the weekend.
Worries are also persisting over whether Greece will be able to meet upcoming loan repayments.
The FTSE 100 closed 82.73 points lower at 6,948.99. Among the banks, RBS dropped 2.8% and Barclays fell 1.9%, while Lloyds shed 1.3%.
However, Royal Mail was having a better day, with its shares jumping nearly 3.6% to 521.50p following a broker upgrade.
Cantor Fitzgerald upgraded its rating on Royal Mail to "hold" from "sell", and also increased its target share price to 500p from 440p.
Easyjet shares rose 1.6% after Irish rival Ryanair reported a big jump in full-year profits, helped by rising passenger numbers and lower fuel costs.
AstraZeneca shares fell 2.2% after Amgen said it was ending a collaboration with AstraZeneca over developing a psoriasis drug.
On the currency markets, the pound fell 0.52% against the dollar to $1.5409, but it rose 0.56% against the euro to €1.4140.
(Close): Worries over prospects for the eurozone weighed on banking shares, helping to drag the FTSE 100 lower.