Big post, here goes...
As Dimlee mentioned, this has been raised before (geddit?). The first thing we need to look at is what are you referring to when you say 'airships'? Rigid or non-rigid? As we know, the US Navy had a sizeable fleet of non-rigids in use in WW2, as the RNAS had a sizeable fleet of them during the Great War - around 200 all told, although not all at the same time. The British had a big airship building scheme, like the Germans, but British use was far more sensible and successful. The use of rigid airships for military use was/is a supreme waste of resources As has been highlighted, airships in general cost enormous amount of resources to build and operate. In WW1, to build an airship shed large enough to house a big rigid cost about the same as building a destroyer and used more metal. Airship stations were vast and had enormous number of personnel. In the UK at RNAS stations, locals villagers were used for repair of airship envelopes, as nursing staff, or as simple labour force. Those big rigid required around 400 people to walk them out of their hangars if there was no mooring mast. So, very labour intensive and very expensive.
A typical RNAS airship station with two Coastal sheds (for Coastal Class non-rigids) and a big shed, over 700 ft long. Note the hydrogen processing plant at lower right.
EF airship facilities
The loss of airship expertise in the 20s and 30s guaranteed that Britain would not restart its military airship effort after the loss of R.38 (built by Short Brothers at Cardington for the US Navy) and since Germany was not allowed to use airships for military purposes and it was cheaper to built aeroplanes for its Luftwaffe in the 30s (Goring didn't like them - he ordered Graf Zeppelin and Graf Zeppelin II scrapped in 1940), it meant the US Navy was the only country with the resources available to operate them.
On to policy behind their use. The primary role, maritime patrol was probably the best use of airships in warfare and their success is difficult to quantify depending on what you use to measure it. During WW1 the RNAS made the claim that no convoy escorted by airships was successfully attacked by submarine, not true - there was one example - but the effect of having an airship nearby forced the submarines to submerge and remain so. The deterrent factor was enormous and there are accounts of Uboat crews having admitted their escape owing to the presence of airships. Lets recall that submarines in WW1 and in WW2 were slow and unmanoeuvrable underwater, unlike today - their normal speed submerged was between 4 to 6 knots. Non-rigid airships, while not as fast as fixed wing aeroplanes, were quick enough, they could also come to a halt in the air and in WW1 in particular they could carry a sizeable warload and radio equipment, which not all maritime patrol aircraft were equipped with - by the time of WW2 radio equipment had shrunk in size but increased in complexity.
The non-rigid SSZ.59 about to land on the deck of the aircraft carrier HMS Furious in 1918. The Submarine Scout Class ships were quite successful, being small, manoeuvrable and reliable, with a RR Hawk engine that never missed a beat.
SSZ 59
The thing is, by WW2, aeroplanes had improved beyond the capabilities of airships; and while airships carried out patrols lasting as long as 48 hours (the limits being in crew comfort and facilities), aeroplanes were cheaper to build and more reliable and by WW2 could carry big warloads, were faster and better defended than airships. Airships were an expensive luxury borne from necessity and a lack of performance in aeroplanes of the day. Once aeroplanes became better at what they could do, there was no place for airships.
R.29, the only rigid airship to have taken part in the sinking of an enemy submarine, UB 115 in October 1918.
R29 s
As for the lifting gas, the US Navy used helium in its airships, but three of its four big helium rigid airships were lost in accidents - all weather related, so that alone tells you that the dangers of hydrogen were relative (the RNAS had very few hydrogen related disasters - it was down to quality control) and that weather was the bigger killer of airships.