The Brexit Age Gap
As part of a speech to activists in Southport, Liberal Democrat leader Sir Vince Cable said:
75% of under 25s voted to Remain. But 70% of over 65s voted for Brexit.

It is clear that age was a factor in how people voted in the 2016 EU membership referendum. An analysis of wards showed a marked relationship between the average adult age (from the 2011 census) and the Leave vote share.

In the same analysis, there was a stronger relationship found between educational attainment and voting Leave:

This article focuses on how big the difference between age groups is, and in particular, if “70% of over-65s voted for Brexit”.
British Social Attitudes
Sir Vince Cable’s source for his claim was not stated. Since it was a private ballot, we are reliant on estimates from surveys asking either about vote intention before the referendum or vote recall after the referendum.
The highest quality data would come from a random probability sample of the British public undertaken after the referendum. Such a survey exists: the British Social Attitudes survey, mainly carried out between July and October 2016.
According to the BSA survey, the central estimate was that 72% of voters aged under 25 voted Remain, whilst 63% of voters aged over 65 cast their ballots for Leave.

The BSA chapter on the EU referendum also finds an educational difference, where graduates were more likely to vote Remain. The chapter states:
The pattern of voting in the EU referendum reflected then, above all, an educational divide. At one end of the spectrum most graduates voted to remain in the EU — at the other, most with few, if any educational, qualifications voted to leave.
Other Surveys
There are other surveys to consider.
The poll conducted on referendum day, on behalf of Lord Ashcroft polls, found an estimate of 73% of voters aged under 25 backed Remain and 60% of voters aged over 65 supported Leave.

However, Ipsos MORI split the older age group, estimating that 66% of voters aged between 65 and 74 backed Leave, whilst 63% of those aged 75 or over did so.
Verdict
Given the range of survey estimates, it is probably not true that “70% of over-65s voted for Brexit”.
Moreover, whilst Leave had “overwhelming been the choice of the older generation”, education mattered more than age: whether looking at surveys or analyses of real vote patterns.