In general, insurance companies can raise your rates due to an at fault accident or after an evaluation of your claims history but cannot go up just to cover the cost of repairing your vehicle. The amount of your claim may be taken into account but your rates should not go up by that same amount just to cover those expenses.
The amount of an auto insurance increase in premiums after an accident will depend on a variety of factors such as state law and your insurance company rating plan so it is difficult to give an actual amount that it would go up. If your insurance rates will go up and if so by how much will depend upon the rating system that your insurance carrier has in place. These rates are filed with your state's insurance regulator and regulated by your state's insurance laws.
Besides these first 2 factors, a rise in rates can also depend on the nature and severity of your vehicular accident and who is found to be at fault. A minor accident, like yours of hitting the curb and not anyone else's property, may be forgiven under state law or by an insurance company. Some companies have a "forgive the first accident" policy but the qualifications for this and if this is available differ from insurance company to insurance company.
If you have had a good record, and you have been with a particular company for more than five years, there are some car insurance companies that will not increase your rates if you are found to be at-fault in a relatively minor vehicular accident. If instead you were involved in something serious like street racing and had your accident with the curb, you would likely see a bigger increase in your insurance premiums. So an insurance company takes into account if your accident was minor or major.
It would appear that your accident with the curb was minor and thus it likely would not make a difference in your rates whether your repairs were $700 or $2500 but the type of accident that your insurance company will look at. Since insurance laws vary by state and insurance rating systems differ from one insurance company to the next though you will need to ask your insurance agent how specifically your rates will be affected by this incident and claim by discussing your insurance provider's rating system.
Typically, it makes more sense to file the claim regardless of the future impact on rates. The impact usually isn't greater than the cost. And after all, that is the reason you buy insurance, to be protected.
From the information you receive from your agent you should be able to determine if you want to make a physical damage claim with your insurance carrier or pay for the damages on your own.