How can I get my husband criminal record?
You can go to a website like intelius.com and pay a relatively small fee for a background check that you can do right online. You can also contact a private investigator in your area who will run this check for you. It will cost a bit more than doing it yourself, but is still probably going to be a fairly reasonable rate.
Can you cross the border with a criminal conviction?
Yes, you can cross the border for a limited-time visit. The border guards do not check criminal records for people simply crossing the border unless there is a reason to suspect the person (of potential terrorism or something similar). However, you will likely be rejected if you attempt to immigrate into Canada (a criminal background check is a routine part of that process).