Many credit cards have no annual fee and some welcome bonuses. For example, after one spends over a certain amount in the first say three months, the credit card user gets $200 reward.
I am thinking about applying for such a credit card, using it only to get the welcome bonus, and then either leaving it unused forever, or simply closing it after paying off all the balances.
Is this a good idea, or does it have any problems?
That's one option and there is nothing wrong with doing that. Please carefully read the "fine print" so that you understand what your responsibilities are. Sometimes there are minimum usage requirements and the like.
As long as you are happy with what you have to do to get and keep the $200, then feel free to apply.
This may have a negative impact on your credit score. The exact amount would be difficult to calculate.
A medium-weight factor in calculating your credit score is the average age of active accounts. Opening a new account will reduce this average.
A low-weight factor in your credit score is whether you have any recent applications for credit. As implied by AxiomaticNexus's comment on your question, merely applying for this credit will also reduce your score by at least a few points.
A high-weight factor in your credit score is your utilization ratio. If the limit on the new card is high enough and your usage is relatively low, this might outweigh the other factors.
Personally, I already have a relatively high credit limit with pretty old accounts, so adding another credit line hits me harder than someone with a low credit limit or newer accounts.
