Making Infantry Obsolescent
Most Star Fleet larger than shuttles contained four weapons that obviated infantry organizations.
The Photon Torpedo - this was an matter/antimatter weapon that could contain up to several kilograms of antimatter explosive. Useful against similar-class targets (other starships). Against grouped bodies of enemies, or in cases of total war, these could also be used with devastating effect against ground targets. Modern Equivalent: onboard fission weapons
Phasers - this phased energy weapon was for cutting through shields and armor and providing more focused offensive power. Modern Equivalent: standoff airborne cannons (like the 30mm gun on a C-130), or standoff naval cannons (like the Zumwalt main gun)
Sensors - these sensors could pick out smaller-than-human life signs from orbit, in 3-dimensional stacks (cities), and sometimes several meters underground. It could pick up weapons, weapons fire, and other minute details. Modern Equivalent: satellite sensor suites and global communications.
Transporter - combined with sensors, the Star Fleet ship could put a smaller security force where it would do the most good. Maybe after softening up a target with phasers and photon torpedoes. Conversely, the transporter could be used to directly sweep up a belligerent leader from the field and made to answer for his/her actions. Modern Equivalent: satellite + airborne sensor suites, high-altitude/low-opening troop drop, and VTOL pickup.
TL;DR : a "contemporary" Star Fleet would consist of no infantry. Most large vessels would contain at least a few nuclear weapons with doctrine allowing these weapons to be used in situations ranging from ship-to-ship engagements, to softening up ground targets. The ships would be connected to a satellite sensor network, and have at least some small aircraft capacity for landing security forces.
Making Airforces Obsolescent
Compared to small craft, Star Fleet vessels had enough size, speed, and precision that even swarms of small craft of comparable technology (Star Fleet shuttles engaging misbehaving star ships) could be batted away easily.
Speed - a Star Fleet vessel usually had far superior warp capability to small craft. The captain of a Star Fleet craft could usually choose to disengage at-will. Modern Equivalent: with a liquid-lead nuclear reactor, the soviet Alfa class submarine could move as fast underwater as any surface ship on the planet, quickly submerge to disengage, and outmatch any aircraft for endurance.
Precision - again, phasers with short linger time on targets allow big Star Fleet craft to mop up large numbers of smaller craft. Modern Equivalent: while the LAWS weapon system only has an engagement range of 1 mile, it might be the best fit for engaging massed smaller craft.
TL;DR subsurface capabilities and a liquid lead nuclear reactor provide the ability to put aircraft out of range. Combined with a laser or other good anti-aircraft system, the effectiveness of air forces would be significantly less.