Two militants were killed in the exchange of fire with security forces, a police spokesman said.
BSF officials said an ASI of the force was killed in the attack on the 182nd Battalion camp.
Two BSF jawans and a policeman were also injured in the attack, the officials said.
The attack took place at 4.00 am and the exchange of fire is still on between the militants and security forces, they said.
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Outbound passengers have been now allowed to proceed towards the airport where operations were initially suspended as security forces engaged militants who breached the 182 battalion headquarters of the BSF located in Gogoland, police said.
This battalion is entrusted with the security of the runway of the Srinagar Airport.
Adjacent to the camp is the old Srinagar air field which is been run by the Indian Air force.
The area also houses training centres of the BSF and the CRPF.
Schools in the vicinity of the airport have been closed.