I can think of three ways to do this. Before you do any of these please make sure that you make a backup of your database. Please fully read all three before you attempt any of them. All of these require manipulating your database at the database level utilizing a 3rd party piece of software such as but not limited to MySQL Workbench, MySQL Admin, Navicat, etc.
1. truncate the table, this will erase all the data in the table however.
TRUNCATE TABLE ost_ticket
Also keep in mind that doing that will break the relationships between any tickets and the messages so chances are you would also want to do that to ost_ticket_attachment, ost_ticket_message, ost_ticket_note, and ost_ticket_response.
2. run a MySQL query to reset it. If you have data in the table that you want to keep you shouldn't do this because it will result in multiple duplicate ids. (note: that would be bad)
ALTER TABLE ost_ticket AUTO_INCREMENT = 1;
3. Lastly and probably the safest (which also translates into the most effort) is to do selective backups of each table. Wipe out the database schema entirely. Then browse to OSTICKETDIR/setup/inc and locate the osticket-v1.6.sql and import it. This will give you a empty default OST database. Then restore the tables that have data that you care about. Most likely this list will consist of ost_api_key, ost_config, ost_department, ost_email, ost_email_banlist, ost_email_template, ost_groups, ost_help_topic, ost_kb_premade, ost_staff, ost_syslog, ost_ticket_priority, and ost_timezone.