What do manufacturers do to reduce cabin noise?
it is being claimed by the Airbus that their new planes offer more sound proofing than those of Boeing


I guess you mean this sort of claim:


  No matter where they sit, all passengers benefit from the quiet A380 cabin with on average half the sound energy of other aircraft. Passengers' feedback confirms the extraordinary contribution to their general well-being due to the absence of background noise on the A380.
  
  In market research surveys the A320 is regularly rated higher than its rivals for the low level of noise in its cabin. Effective insulation in the cabin is one reason why passengers say they find the A320 a quiet aircraft in which to fly. The A320 cabin is the subject of continuous improvements and in its latest version, the noise has been reduced by a further 1dB.





  how do Airplane manufacturers tackle the problem of soundproofing the cabin


There are a variety of methods


soundproofing materials in the wall of the fuselage between inner and outer skins.
choice of interior materials and furnishings etc. 
adjusting cabin interior contours to allow lower airflow velocity.
fuselage aerodynamics.
engine nacelle accoustic liners.
scalloped edges on engine exhaust nozzle.
larger slower engine fans.
sharklets.
etc


BBC report


  One of the main sources of noise is wind. So during the design phase of modern aircraft, computational tools model the aerodynamics of the aeroplane to highlight areas of high airflow that are likely to increase cabin noise. This allows engineers to reshape their designs, says Alan Pardoe, director of product marketing at Airbus. Such techniques, he says, helped the double-decker Airbus A380 plane to be recognised by the UK Noise Abatement Society as the quietest jet airliner on the market.


Airbus say


  When it comes to making the A380 one of the quietest aircraft in the sky, it's the small details that count... in fact, more than 3 million of them!
  
  This is the number of tiny perforations in acoustic linings on the large nacelles that surround and protect the A380’s jet engines.  The acoustic material helps dampen noise created by the airliner’s four turbofan powerplants, ensuring the A380 is a friendly neighbour wherever it operates in the world.
  The A380’s nacelles are supplied by Aircelle, which has worked closely with Airbus in minimising the jetliner’s noise output during takeoff and landing.  Acoustic panels are applied throughout the engine – from the air inlet to the exhaust nozzle.
  
  Each panel, which is produced from composite material or titanium, is designed to attenuate noise at different frequencies, further improving the soundproofing effect.  Noise-reducing innovations include Aircelle’s 2-degree-of-freedom acoustic panels, which use a stack of two honeycomb composite sandwich layers separated by a porous wall.
  
  These panels dampen high-frequency noise from the engine turbines.  With much of this noise source successfully dampened by the acoustic treatment, engineers and designers are now focusing their attention on treating other sources of aircraft noise, promising even quieter jetliners for the future.


and also this


  The Airbus-developed Zero-Splice engine nacelle liner was designed to reduce the noise generated by the jet engines’ large fan, which represents almost half the noise during both approach and take-off. Following extensive studies carried out by Airbus on how to reduce fan noise, it was found that splices used to join the engine nacelle inlet’s acoustic panels generated forward fan noise – which was counter-efficient to the liner’s acoustic treatment. Airbus therefore developed the Zero-Splice nacelle liner as a single 360 deg. piece of composite, which is comparable in size to the diameter of an A320 fuselage. 


It is, I think, obvious that measures that reduce engine noise on the ground also reduce it in the cabin.

They also write about how wing sharklets reduce noise.

I'd guess there are many other measures and design choices that are introduced to reduce noise.




  What proportion of the total aircraft development cost ...


I'd be surprised if Airbus or Boeing make public any breakdown of their development costs. 

You can find general breakdowns like the following but I havent found any that break out soundproofing development costs.