Is it safe to clear down the msdb..sysbackuphistory table in SQL Server as it is filling up the database?

A. Yes, it is safe. There is a stored procedure to clear up the sysbackuphistory and sysbackupdetail tables available with SQL 6.5 - sp_sysbackuphistory_limiter. This is run with a single parameter of the latest date you want to keep. Everything before this is deleted.

See Q152354 for more details.