What are the ways that engine manufacturers currently minimize the damage caused to engines in a bird strike?
Placement on aircraft that fly above birds. This is the #1 reason we don't hear about airliner bird strikes every day on the news; cruise altitude for airliners is around 30,000 feet, while only two species of bird have ever been recorded flying at that height, one in Central Africa, the other around the Himalayas, neither of these being common travel lanes for airliners. The critical time for most bird strikes is below the cloud ceiling and/or transition altitude, above which the air becomes thinner and birds have to work harder to stay aloft. Thus most birds fly below transition, while airliners spend as much of their flight time as feasible above it as the thinner air reduces drag and lowers fuel consumption (the "strata" of thicker air at the transition point, in addition to the simple reduction of SPL on the square of distance, also helps insulate people on the ground from jet noise).
Strong intake fan fins. The first set of turbine fins that you see looking into a nacelle are the ones in the direct line of fire. They're also the largest fins, but in their favor there are higher acceptable tolerances for those fins versus the ones in the compressor stages. So, they're built strong to hopefully survive ingesting your average goose. Here's a YouTube of the A380 engine's bird strike test; the engine continues to operate even though the blade that was struck is misaligned on the turbine hub for a few rotations.
Garbage disposer effect. At ascent throttle levels, an airliner's engines are rotating between 10,000 and 15,000 RPM. Even travelling at cruising speeds of 500-550 knots, the rotation speed of the turbine will chop most organic matter into mincemeat in a fraction of a second. In fact, bird strike tests show the bird carcasses literally exploding from the force of impact with the rotating blades. These smaller pieces cause fewer problems passing through the engine.
High bypass. About 80% of the air volume entering most underwing airliner engine nacelles passes around the turbine chamber instead of through it, increasing thrust while reducing fuel consumption. Fuselage-mounted engines on older aircraft had lower bypass ratios (one of the reasons those designs are being phased out of modern fleets). A bird entering the engine has a similar chance to pass right on through without encountering the combustion chamber.
Reinforced nacelle housing. In the event a bird strike does dislodge something in the engine, the nacelle housing is engineered specifically to contain the shrapnel, minimizing the chance of damage to fuel and control lines in the wings and of course to the fuselage. (Here's a YouTube of a "blade-out test" for the A380's Rolls-Royce engines) The pilots will be alerted the engine has failed, and will cut off the fuel supply, engage fire suppression and run on the remaining engine(s).
Overspecification of required engine thrust. One of the major tests an airliner design must pass before being FAA certified is an "engine cutout at decision" test. An aircraft is loaded to MTOW, taxis to the runway, the engines are spun up, the aircraft rolls out, then at the "decision speed" above which the pilot must commit to a takeoff, an engine is idled. The aircraft must still be able to get off the runway and attain a minimum climb rate. That test, for most modern two-engine jets, shows that the plane is perfectly capable of flying on one engine (if such a thing happened at takeoff in real life the jet would circle and land again but it would not have to risk an aborted takeoff above decision speed).
In short, these engines are built tough and tested tough, as is the rest of the aircraft (manufacturers build several planes' worth of components and subsystems for the express purpose of torture testing, and that's if they get it all right the first time). In addition to bird strikes, engines are tested for water ingestion, hail strikes, sand ingestion, smoke ingestion (a British Airways 747 lost all four engines after flying through the upper ejecta cloud of an erupting volcano; they got two restarted and landed safely, but future engine tests got even more stringent to avoid a repeat performance), and a host of other things that just aren't good for jet engines.
The most famous bird strike incident in recent memory, Sully Sullenberger's emergency water landing in the Hudson River, was caused by hitting not one or two, but an entire flock of birds, and not sparrows or pigeons, but Canada geese, which can weigh up to 20 pounds each. This incident is still a good example of how well engineered these engines are; both engines would have been at full revs as the plane was climbing out of LaGuardia, yet AFAIK the turbines, though deformed, didn't even part company with the hub, saving the control surfaces and lines for the ditch attempt, resulting in no fatalities and only one hospitalization.