Have a gutter spout that drains right on sidewalk near front door.

you can buy extension pieces to change the direction of a down spout or french drains that attach to the down spout and can be re-routed and even buried in the yard

One of my neighbors used PVC pipe as an extension to redirect the rain water. It is neater than adding the usual extension pipes and can be run up against the house to a better drainage area. It could even be put under ground if you aren't concerned about it clogging. We have an underground drain that runs from our home almost to the road. There is enough of a drop for the water to flow. If your sidewalk is poured cement and pretty sturdy you can run the pipe under it. You just need to try not to dig so much soil out that the walk shifts and be sure to repack the ground as much as possible. If the walk isn't solid just dig it up and run the pipe. In Florida, where the ground is mostly sand, we used water pressure from our garden hose to create a tunnel under our driveway to run a narrow pvc pipe for our watering system. Depending on your soil there are lots of options.

You can reroute the drain pipe or move it to another location. Removing the drain pipe, sealing the hole, and move it to another spot.

You could have a gutter guy come out and replace that section of gutter, and relocate the downspout. Most handymen can do the job too.

If your up to tackling it on the ladder.... https://www.youtube.com/watch?v=Xtt4CwstgAI