Summary: Your problem is a doozy. No solution is perfect. Murphy's law will get everyone killed. Greed, and laziness are dangerous. Well funded malice is almost impossible to defend against.
Note: I will use Earth and other objects in our Solar System as a short hand. Adjust the details to fit the setting.
Solution 1: Lasers
Lasers are problematic. If a craft is speeding on an intercept course with Earth, and you shoot it with a laser you now have a piece of wreckage on an intercept course with Earth. The laser does not actually change the trajectory of the ship. A mega-powerful laser could possibly explode the thing into a million pieces that would then scatter and expand like buckshot. This would make a lot of smaller craters, which is almost as bad.
Switching from lasers to giant bullets or missiles might be more effective, but it brings up a ton of problems. Your bullets would have to be as big as the craft they are knocking out of the way, and you would have to launch them way early, because they move slower than lasers.
For any solution, your early detection system would have to be incredible. You have to be able to spot objects that are tiny on an astronomical scale while they are at least 10AU away, and when you first see an approaching craft traveling at 0.3c, it is already 30% of the way to you. Any solution that knocks dangerous craft out of the sky is problematic for many reasons. On the other hand, it is the future. I assume you have some great technology.
Another early detection problem is that it would be nearly impossible to tell the difference between a ship that would impact Earth and one that would stop next to Earth. They are on nearly the same trajectory.
I think it can be assumed that the people running these defense systems are grim and humorless. They would be required to blow up people if there was any chance they were a terrorist, or if they were off course headed towards Earth. Zero tolerance for mistakes or malfunctions. Plenty of innocent ships would get killed.
Solution 2: Completely automated space flight.
Completely automating things helps, but machines aren't perfect. Machines designed, installed, and maintained by humans are prone to error.
Human error or equipment breaking down will cause screw ups. Not too long ago, Air France Flight 447 crashed after a air speed sensor iced over. The autopilot disconnected when it sensed an equipment malfunction and the human pilots reacted wrong. If your relativistic flight becomes as common place as air travel, things will go wrong. Murphy's law is about human nature, not technology. You can make the pilots machines, but the technicians who install the machines might still be bozos, or the engineers who designed it might have missed something, or an unexpected solar flare pops up, or any of the other 1000 things that no one has thought of yet.
Solution 3: Strict Licensing and Regulation
Regulation is not a perfect solution. If you put super strict regulations on modern airlines, you would have an entire industry looking for ways to cut corners. The airlines are in a bitter cost cutting war right now. Some of them would try to evade the regulators, because that could be cheaper than following the regulations. If you sent an army of regulators, some of them would take bribes. If you sent an army of unbribable regulators, the industry would buy lobbyists. A small number of very rich, highly motivated transportation executives could inadvertently destroy your fictional Earth.
Nuclear power presents a good counter-example to air travel. The potential for disaster is there, but it so far hasn't happened outside of Russia. Everything went wrong in Fukashima, but the damage was contained with minimal loss of life. There are fewer people involved in nuclear power than the airline industry, the profit margins are not as thin, and the government regulation is a lot stricter. Nuclear power spends a lot of time and money on a system of layered fail safes to keep giant disasters at bay.
These safe guards are expensive and time consuming. We have them because the accidents we did have made us believe that nuclear power is scary stuff. If the Earth had a "minor" accident in your universe, it would be more plausible for people to take extreme precautions. Maybe the kind of minor accident that just destroy's a couple of states, not a continent. If there are no near misses, people eventually let their guard down. Conversely, if the public is terrified of something, no amount of lobbying can relax those regulations.
It also helps that there are relatively few nuclear power plants. We live in an age of air travel where airlines run thousands of flight per day while trying to cut costs to the bone. If you are imagining an equivalent system of relativistic rockets, Earth is doomed. If there are a handful of giant flights per week, we might do ok.
A lot of my discussion has been under the assumption that you would have private industry running the show, and government regulating it from outside. It could be a government run program in the first place. Then all the concerns about lobbying and regulation get a lot weirder. If it is a government program that is making a ton of money on ticket sales, you get to have all of the problems with greedy private industry AND all the problems with hidebound government bureaucracy.
Solution 4: Force fields
Handwavium powered force fields can solve any problem ever. I assume yours will combine nano-technology with applied phlebotinum. I don't think there is a hard science way to do this. I could be wrong.
New Solution: Inbound speed limits
If you are headed towards Earth and closer than Jupiter you have to limit your speed way down. This makes the giant bullets proposed as an alternative to the lasers far more effective. This also gives defense stations a little more warning for malicious attacks. When someone doesn't slow down after passing Jupiter, blow them up.
New Solution: Be VERY VERY nice to anyone powerful outside the system.
If an enemy government outside the system wants to destroy Earth they can. Simply launch a field of gravel at Earth at relativistic speeds. It could be hard to detect, and even harder to shoot down. The lasers or massive bullets would take out one pebble each. The only defense here is your handwavium powered force fields, and I imagine they can be overwhelmed. The pebbles would slam into the Earth's atmosphere so hard they would cause fusion explosions. That's what happens when something travels at relativistic speeds in an atmosphere.
New Solution: Gravel Launchers and Diplomacy
Build your own gravel launchers and make it clear to the space Soviets, that you will destroy them if your own planet gets destroyed. Mutually Assured Destruction.
New Solution: Off planet terminals
All relativistic travel must slow to a stop at the L4 and L5 Earth Sun Lagrange points. These are stable orbital points about 1AU from Earth. They are a nice place to park an orbiting bus station. The bus station is much smaller than a planet, therefore it is less likely to get hit on accident. It also doesn't have 10 billion people living on it. Any ship seen not heading for a bus station, will be destroyed. The distance from Earth means that it is much easier to tell if a ship is heading towards them or towards Earth. Therefore you have more time to deal with rogue ships.
