The 2010 AJL survey reported the following results: 80.2% of synagogue libraries are automated, using a variety of systems
The most popular systems (in descending order) are:
ResourceMate (note that ResourceMate supports both Hebrew and English scripts)
Follett
LibraryWorld
Alexandria

A free (or close to free) system is called LibraryThing. Billed as "the world's largest book club" it can be adapted to the needs of a small library.

Surpass CL is an automation system "streamlined for churches, synagogues, and other places of worship."

A smaller number of synagogue libraries use the following systems:
Agron
Innovative
OPALS (note that OPALS supports both Hebrew and English scripts)
InfoCentre from Sagebrush is used by a number of synagogue and center libraries.
Sagebrush is now owned by the Follett Corporation.
Follett Software Co. continues to provide support for current users of InfoCentre but it is no longer being sold.

Additional Resources:

How to automate a library one step at a time, 2014
Planning for Automation and Use of New Technology in Libraries
Top Library Automation for Software Products
Automating Libraries: a selective annotated bibliography from ALA
Leff, Barbara Y.: Planning to Automate Your Synagogue, School, orCenter Library? (Included in appendices are a List of Selected Library Management Software for Small Libraries by Susan Dubin and Librarycom.com Overview by Cindy Gerecht.) --Presented at the 2001 AJL Convention.
Landau, Rebecca: Bar Coding a Library: Issues and Concerns (poster session) --presented at the 2001 AJL Convention.