Is there a way to sync up Messaging on my iPhone and MacBook Air?

Greetings hopealways, 

Be sure that you have Messages enabled for iCloud. The following article goes over how to set that up and what to expect when using it. Keep in mind this is not a backup for messages and if deleted, the messages cannot be recovered. Use Messages in iCloud

Here's how it works

When you turn on Messages in iCloud, your message history is updated automatically on all of your devices. That way, you always have the same view everywhere you use iMessage. When you delete a message, photo, or conversation on one device, it’s removed from all of your devices. And since all of your attachments are stored in iCloud, you can save space on your device.

You can use Messages in iCloud on your iPhone, iPad, iPod touch, Apple Watch, and Mac. For your privacy, Messages in iCloud is end-to-end encrypted, which means you can't view or access Messages online via browser. 

Thanks.