Whether this is a good idea or not really depends on how active you are with your money management or how lazy you are about it. If you are careful with your credit card spending, and always keep within your budget, automatic payment can reduce the time you need to spend managing your credit card account and paying bills.
If you occasionally splurge and need to keep a balance on your credit card (usually a very expensive form of credit) then paying off in full could lead to more money being taken from your current/checking account than you might expect, which could cause other debits to be refused.
However, depending on where you are located, you may have another option.
In the UK, most banks offer the option to pay off the minimum amount automatically by direct debit, rather than paying off the full amount. For instance:
You must pay us £5 or 2.5%, whichever is higher, of your statement balance, or if less than £5, your statement balance.
Paying at least this portion automatically can be very useful if your credit card company charge you a late payment fee. E.g.:
Default charges, Late payment, £12
In combination with a standing order, it can also make budgeting easier. You guarantee to pay off the minimum charges every month using the minimum direct debit, and then pay off an affordable amount every month to cover interest and reducing your principal debt using the standing order. This gives you finer control, but also ensures that you are never stung with late payment charges.