Support for the strike was lower than seen in an earlier ballot, which had a higher turnout and which saw 92.4% of those voting backing a strike. However that vote, which would have led to stoppages over Christmas, was deemed illegal by the High Court.
Legally, the union must give a week's notice if it plans to take action - meaning strikes could begin from the first week of March.
Unite has already pledged that there will not be stoppages over the Easter period.
BA has threatened to take away generous travel perks for workers who do strike.
The vote result comes as passengers are suffering delays and cancellations due to industrial disputes elsewhere in Europe.
On Monday, 4,000 pilots at German airline Lufthansa began four days of strike action.
The carrier has cancelled about 3,000 flights and has warned of delays both domestically and internationally.
Meanwhile unions representing French air traffic controllers have announced a four-day strike, beginning on Tuesday, that will result in hundreds of flights in and out of Paris' Charles de Gaulle and Orly airports being cancelled.
Earlier this month, British Airways announced it made a pre-tax loss of £50m ($79m) in the three months to December 2009, down from the £122m it lost a year earlier.
The changes which prompted the strike ballots were part of cost-cutting measures at the airline.
In November, BA reduced the number of cabin crew on long haul flights from 15 to 14 and brought in a two-year pay freeze from 2010.
The airline has also proposed new contracts for fresh recruits and newly-promoted staff. These include a single on-board management grade, no seniority, promotion on merit, and pay set at the market rate plus 10%.
This would still see new recruits paid significantly less than current staff, however.
On Friday, a High Court judge ruled that BA was within its rights to make the changes to save money.
The union had claimed it was not consulted properly, but the judge said the "less-than-extreme changes" including freezing pay and cutting crew on long-haul flights, were not unreasonable.
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