It looks like the main water pipe is leaking under my house. The water comes into the house under the basement - I can see the water comes before water meter in the whole in the basement and it goes out of the house in water sewer. I'd like to know if I am responsible for the leak - I believe it is part of street main water pipe - and I'd like to know the price. Thank you!

Usually, you are responsible from the "curb stop" or where the water is turned off at the curb.

KISS is bang on! The water service that runs from the property line at the curb to your house is your responsibility.
How much it will cost to replace will depend on the depth of the line, it's size and the length of the run. Perhaps this is a simple leak that can be repaired instead of having to replace the service. Have you called a plumber and checked? Good luck, Tom