In the past a US citizen could not use a K1 Fiancée visa to take their Thai fiancé back to the US under a same sex relationship. You would not be able to use a K1 visa, K3 visa or a CR1 visa to take your partner back to the US. Same sex marriages in Thailand are still not recognised by law however without the ability for a K3 spouse visa or CR1 spouse visa you could technically still make use of the K1 fiancée visa.
On June 26, 2013 the United States Supreme Court ruled that DOMA (Defence of Marriage Act) Section 3 was unconstitutional. In terms of the ruling it should be possible to apply for and obtain a K1 fiancée visa for a same sex couple however it had not been tested. This is mainly because there is still confusion pertained to foreign same sex fiancés. The Supreme Court ruling only covered US based marriages in this regard. The United States Citizenship and Immigration Service stepped in and clarified the situation that it was now possible for a US citizen to petition for a K1 visa for a same-sex fiancé in much the same manner as a different-sex fiancé. To the best of my knowledge this has not been tested in Thailand as yet in a K1 visa application.
Same Sex Marriages
The financial requirements for the visa for a same-sex couple would be the same as well as the procedures for different-sex couples. If you are considering taking your Thai partner back to the US then you can apply for a K1 visa in Thailand. You would need to speak to an immigration lawyer in Thailand about this as very few law firms are at the cutting edge of legal changes in Thailand.
Note that as stated above you cannot obtain a K3 visa or CR1 visa in Thailand for a same sex marriage as ‘gay marriage’ or ‘same-sex marriages’ in Thailand is illegal and not recognised. If you partner has children you should also be able to apply for a K2 visa for any minor children. As stated before, this is all new to Thailand so you can expect delays in the process. All requirements would also have to be met such as a visa interview in Bangkok a criminal record clearance check as well as a medical check. Note also that a visa denial can occur so it is best to take proper legal advice on the subject in Bangkok.
Once Thailand does recognise same sex marriages the visa process would become much easier and also open more options for visa applications to the United States. The Thai government has delayed many issues over the past year and LGBT rights have been on the back burner for a long time.
If you are looking for more information you can see the issue covered here (Gay marriage in Thailand) and also (K1 visa application) and the financial requirements listed on this website. Call or email an attorney in Thailand for the latest changes in the procedures, costs and time frame for a same-sex fiancée visa to the US.