What is the best way to keep my team actively logging time?

You can use the automatic reminders in Harvest. And also make it fun, like a shoutout every Friday and the person with the best on-time-runrate will get a special something.

Have them use a browser plugin that will allow them to more easily manage time. Training and ease of use are the best way to drive adoption. If you have specific tools you use everyday (e.g. JIRA) look to see if Harvest can integrate. This also helps people to remember to use their timers.