The Karen say that for the past four months they have been forced from their homes, which are close to where the government's new capital, Pyinmana, is now based.
Bernard Quah, of the United Nations High Commissioner for Refugees, said the numbers of refugees increased after Myanmar's rulers announced they were moving the capital last year.
Analysts say the government's offensive against the rebels could be linked to the move to Pyinmana, a town much closer to the mountainous regions where the country's ethnic rebel groups are based.
Karen rebels are among a handful of ethnic groups who have been waging a war against the Myanmar government for more than 50 years.
Thai border officials told Reuters that about 600 more Karen were camped at the Salween River, which forms the border between the two nations, in an attempt to see relatives living in two camps in northwest Thailand.
Also on Wednesday,
Human Rights Watch, the New York based rights group, said that Myanmar had placed landmines along the country's border with Thailand to prevent civilians from fleeing.
It called on the UN Security Council to put Myanmar, formerly known as Burma, on its agenda after the council agreed in an April resolution to shield civilians from genocide and other crimes against humanity.
Brad Adams, Asia director of Human Rights Watch, said in a statement: "The UN has committed itself again to protecting civilians at risk, and thousands of Burmese are in urgent need of such help."
Myanmar has been run by a series of military governments since a coup in 1962, and has been strongly criticised internationally for its poor human-rights record.