The parish church (Church of Scotland) has registers dating from 1743. Old Parish Registers (before 1855) are held in the National Records of Scotland in Edinburgh, and copies on microfilm may be consulted in local libraries and in LDS Family History Centres around the world. Later parish registers (after 1855) are often held in the National Records of Scotland as are any records of non-conformist churches in the area (often unfilmed and unindexed, and only available there).
The parish registers available worldwide on microfilm include kirk session accounts for the years 1753-1780 (in part 727/2). More information on kirk sessions and their records can be found in the Church Records section of the Berwickshire page.
The LDS Family History Library catalogue lists a microfilm version of Kirk sessions records, proclamations of marriage, 1856-1858, Ayton. The catalogue entry (under Scotland/Berwick/Ayton/Church Records) records this as a microfilm of records held at the Scottish Record Office (now called the National Records of Scotland). The microfilm copy in the LDS catalogue should hopefully be viewable at LDS family history centres around the world.
The Scottish Genealogy Society has published "Berwickshire Miscellany - Ayton & Chirnside" containing Ayton Communion Certificates 1829-1848, Ayton Communion Certificates received 1830-1839, Chirnside Male Heads of Families 1834-1837, and Chirnside Register of Interments 1817-1841.