Selkirkshire Antiquarian Society has published a CD containing transcripts and digital photographs of gravestones in Ashkirk churchyard.
A list of gravestone inscriptions in Ashkirk churchyard may be found in the following volume:
Gravestone inscriptions prior to 1855: Vol 1. Selkirk, Ashkirk and Lindean Old Churchyards
published by Selkirkshire Antiquarian Society, 1969 (listed in the LDS Family History Library catalogue in microfilm format, so hopefully available worldwide in LDS family history centres).
Note: Lindean cemetery lies within what is now Galashiels parish.
A list was also printed in the 1962 volume of the Hawick Archaeological Society's Transactions (it may be an identical list) and a list of the surnames recorded is available online.
Finally, the Scottish Genealogy Society has a list of pre-1855 inscriptions in the parish among its collection of monumental inscription listings. This may be viewed at its library in Edinburgh.