With the opening of the Hokkaido Shinkansen in 2016, it has been easy and convenient to access the city of Hakodate via train from Honshu. The fare from Shin Aomori to Shin Hakodate Hokuto Station will set you back 7,520 JPY at the time of writing, but there is a cheaper way to transit between the two cities if you're on a budget. The Tsugaru Kaikyu Ferry runs 8 times a day from Aomori to Hakodate and vice versa (the journey takes 3 hours and 40 minutes), and the standard fare is priced at 2,660 JPY to 3,650 JPY depending on the time of year.
My family opted to take the train one way and the ferry on the return leg several years ago, and I actually preferred the ferry. It's a great option if you have younger children traveling with you, as you can book suite rooms – this is at a higher fee than the standard fare, but we found it worth it for the opportunity to sprawl out and truly relax on the journey.
Other benefits of the ferry include the ability to bring your canine companion or your vehicle (additional fees apply), the latter of which is extremely convenient if you plan on doing some road tripping at your destination. Up-to-date prices for those additional services can be found on the Tsugaru Kaikyo Ferry website.
If you don't mind the slower speed of the ferry, it can be an economical option to get from Aomori to Hakodate (or vice versa) if you choose the standard seats.