The PCS said 57.2% of those who voted backed strike action – the turnout was 20%.
Although dates have not been set, there is the prospect of strikes affecting the London 2012 Olympic Games because Border Agency staff are involved.
Immigration Minister Damian Green said a strike was “completely unacceptable”.
Details of strike dates and what form the industrial action will take are expected on Thursday. Just over 75% of members who voted were in favour of action short of strike action.
The union said it was in dispute over longstanding issues with the Home Office, including cuts to UK Border Agency staff.
About 16,000 union members were balloted across the Home Office, including in the Border Agency, the Identity and Passport Service and the Criminal Records Bureau.
The PCS said the cuts at the UK Border Agency were continuing “to cause chaos at the borders and queues at airports”.
‘Undermining public services’
PCS general secretary Mark Serwotka said: “We believe [ministers] have acted recklessly and irresponsibly in cutting so many jobs and, in the case of UKBA, they have simply tried to paper over the cracks by deploying severely undertrained staff at our borders.
“If these issues are not resolved, they threaten to seriously undermine the Home Office’s ability to provide vital public services, and we cannot sit back and allow that to happen.”
This week the National Audit Office said the UK Border Agency had laid off 1,000 more staff than intended and was having to hire extra people and increase overtime to meet its workload.
The union said for its members the wider issues under dispute were: job losses, particularly compulsory redundancies; pay and conditions such as government plans to cap pay increases at 1% for the next two years; and privatisation, such as using private companies to clear up a backlog of immigration and asylum casework in the Border Agency.
Mr Green warned the union that he did not believe the public would support any disruption, and called on it to reconsider.
He said: “Only about one in 10 PCS members voted for strike action. The union leadership has no authority to call disruptive strikes on that basis and should think again.
“The security of the UK border is of the utmost importance and we will use our trained pool of contingency staff to ensure we minimise any disruption caused by planned union action.
“Any action that disrupts the Olympics will be completely unacceptable and the public will not support it.”
The PCS is one of the largest unions in the UK with around 250,000 public sector members.
PCS members at the Department for Transport have been taking industrial action over the past few weeks, while staff in other departments, including the ministries of defence and justice, are set to vote shortly on how to campaign against cuts.
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