I'm assuming this pipe connects your gutters to the cities storm drain system, depending on the type of pipe and the damage it's probably a simple fix.
If the pipe is PVC and the damaged area is not too large, you may be able to cut out the damaged section and reconnect it using a coupling.
![PVC Coupling](https://i.stack.imgur.com/rbYPP.jpg)
If the damaged area is larger (and PVC) you could cut out the damaged area, then patch it in using another length of PVC and two couplings.
If the pipe is not PVC you could use a flexible rubber coupling as @AWMoore suggests, just make sure the one you use is rated to be buried (and is permitted to be buried by your local codes).
![Flexible rubber coupling](https://i.stack.imgur.com/Uvi14.jpg)
If this is a storm drain you may want to contact your local water municipality, as they may be responsible for the maintenance of the drain line (paid for by your taxes).
Most towns/cities will not be too happy with you, if you start cutting up their road. So you'll have to use a method that will be completely on your property.
Dump some dirt
The easiest solution would be to build up the area with a load of dirt. Using a wheelbarrow and a shovel, grab some dirt from another location in your yard. If you can't find a place in your yard to take dirt from, you could always purchase some from a gardening/landscape/home improvement store. Dump the dirt along the edge of the road until your hole becomes a hill, then feather it into the rest of the yard. Plant some grass, and grab one of @dbracey's margaritas and watch it grow.
French drain
A french drain would give the water a place to go.
Start by digging a trench right next to the road about 12" wide, and 12" deep (don't forget to call before you dig, so you don't damage any underground utilities). You want the water to drain away, so you'll want a drop of at least 1/8" for every 1' in length. For example. If the drain will be 20' long, the end of the drain will be 2 1/2" lower than the beginning. Next fill the trench with about 4" of crushed stone, and lay your drain pipe in the middle of the trench. You can use either rigid, or flexible pipe.
![enter image description here](https://i.stack.imgur.com/L9y9f.jpg)
![enter image description here](https://i.stack.imgur.com/rkifMm.jpg)
Fill the trench to about 2" from the top with more crushed stone, then finish filling the trench with topsoil. plant grass, and enjoy.
Concrete gutter
If you're familiar with working with concrete, and you like a more finished look. You could install a gutter, along the side of the road.
![enter image description here](https://i.stack.imgur.com/RMSa9.png)
Just remember, you'll need the gutter to slope so water will run away. It's probably a good idea to install a dry well or other drainage method at the end of the gutter, so the water has a place to go (unless you don't care if you simply move the puddle to your neighbors yard).
Best Answer
I am assuming you mean water is getting into your basement and the sidewalk connects to your foundation. My First step would be to try and seal where the sidewalk touches to foundation. There is probably a crack there where the water is getting in. The best sealant would come in a caulk tube make sure it is flexible and made for concrete. Do not get the cheapest stuff at the store. It may or may not work but for $10 its worth a try. A picture would help us give a better answer.![enter image description here](https://i.stack.imgur.com/JA5Dh.png)