Peter Givens, a gamekeeper from Stow near Galashiels, was fined £300 after pleading guilty at Selkirk Sheriff Court .
The two birds were found dead in a crow trap on the edge of woodland at Cathpair Farm on September 23 last year, with the trap’s identification number registered to Givens.
Crow traps can be used under licence but conditions state they mu be checked at least once every day at intervals of no more than 24 hours.
The birds found in Givens’ trap had been dead for some considerable time, prosecutors said.
Both birds would have died from a combination of dehydration, starvation and exposure to the weather elements, with the barn owl and goshawk enduring considerable suffering.
Fiona Caldwell, Procurator Fiscal for Wildlife and Environment at the Crown Office and Procurator Fiscal Service (COPFS) said: “Peter Givens reckless actions and his failure to release these birds unharmed led to their suffering and deaths.
“Wild birds are given strict protection under our wildlife laws and COPFS will continue to prosecute such cases where appropriate to ensure that offenders are brought to justice.”
Crow traps are used for a number of purposes, including the protection of lambs and to protect bird nests from crows looking for eggs, it is understood.
RSPB Scotland criticised the "very low fine” given in the Givens’ case.
A spokesperson said: “We are disappointed at the very low fine imposed by the court in this case. Two protected birds suffered a long, lingering death because of the actions of this experienced gamekeeper.
"Crow traps such as this are in widespread use in our countryside, permitted by general licences issued annually by NatureScot, but nobody knows how many there are, where they are and what they are catching in terms of target species or by-catch.
"At a time of a biodiversity crisis, such unmonitored and unaccountable killing needs to change.”
A spokesperson for Scottish Gamekeepers Association said it was understood Givens’ believed the trap was deactivated at the time.
The spokesperson said: “This case highlights the importance for anyone using legal traps to make sure they are deactivated and properly put away in accordance with the law.”