You will want to enable auditing on the server which hosts the network share. By doing this it will log all the monitored events into the Windows Security log.
You do this by going to start/programs/administrative tools/Local Security Policies. and then navigating to Security Settings/Local Policies/Audit policy.
You are given the choice to monitor success and/or failure of certain events. For files, I believe you are going to want object access and privlege use.
After that you, if you are using an OS prior to 2003, you will need to turn on auditing at the file or folder level.