The meetings reunite relatives separated for 50 years
South Korea says it strongly regrets North Korea's decision to postpone family reunions which were due to take place next week.
North Korea's state-run media said its decision was based on what it termed the tense atmosphere in South Korea following last month's attacks on the United States.
South Korea has been placed on heightened alert in response to the 11 September attacks.
But South Korean Unification Minister Hong Soon-young, said on Friday the security precautions did not specifically target North Korea.
"They are meant to deal with the general situation in connection with retaliation against terrorism by the international community," he said.
"This issue, therefore, can never become a pretext to affect the implementation of agreements between South and North Korea," he said.
The BBC's Caroline Gluck said North Korea felt personally disturbed by the extra security.
Seoul's Yonhap news agency quoted the broadcasts as carrying a statement from Pyongyang's Committee for the Peaceful Unification of the Fatherland which said that Seoul's security-tightening measure "are dangerous acts that severely incite us".
Fourth reunion
The reunions would have been the fourth such event to be held since the leaders of the two countries met at a landmark summit in June last year.
The reunions were scheduled to be held from 16 to 18 October, and to bring together 100 family members from either side of the border.
Many of them are more than 80 years old.
The reunions helped boost trust between the two Koreas, which remain technically at war after a truce, rather than a peace treaty, brought an end to fighting in the 1950-53 conflict which saw China backing the North while the United States supported the South.
South Korea's Red Cross called for a speedy resolution to the problem.
"The issue of family reunions must be resolved as soon as possible, since it is a humanitarian issue," Lee Byung-woong, special counsel for South-North exchanges at the South Korean Red Cross, told Reuters.
"It is greatly regrettable that this has happened with the date already set for 100 visitors counting the days until their meeting," he said.
Not all cross-border links are being severed. Talks and ministerial discussions will go ahead later this month.