Is the UK suffering from a participation crisis?

Yes – Low turnout at elections undermines the legitimacy of the resultTurnout at the 2015 General Election was 66.1%, far below the high levels of the 1970s, where it reached 78.8% in 1974. Of the 66.1% of the electorate that participated in 2015, only 36.9% voted for the Conservative Party. This means that the current Conservative majority Government was endorsed by just 24.4% of potential voters.  As a representative democracy, it is vital that turnout is high so that the resulting government can claim to have a legitimate mandate to enact the policies in their manifesto.  The 2012 Police and Crime Commissioner elections had record low turnout, with an average of just 15%, undermining the notion of elected and accountable police officials
However – Low turnout has yet to lead to produce an actual ‘crisis’While general election turnout is lower than the highs of the 70s, it has been steadily increasing. Although only 61.4% of the electorate participated in 2005, this increased to 65.1% in 2010, and then 66.1% in 2015, and 2.5% higher in 2017.  Many others might argue that the real crisis is to do with our electoral system. For politically active people living in safe seats, it could arguably make more sense to abstain and participate in other ways.
Yes – Turnout has also been poor in many recent referendums, undermining the legitimacy of the results
It is often argued that referendums and initiatives could be a means to encourage political participation, as citizens might be more inclined to vote when they know that they have the opportunity to make their voice heard on a particular issue. However, many referendums have had turnout that is far below that of recent general elections. Only 34% of eligible voters participated in the 1998 referendum on whether to create a London mayor and London Assembly. Not only did only 50.1% of the Welsh electorate participate in for the 1997 Welsh Assembly referendum, but the ‘yes’ campaign won by the extremely narrow majority of 50.3%. Only a quarter of the electorate endorsed this substantial constitutional reform.
However… some referendums have attracted higher turnout, and further referendums have been requested
The 2014 Scottish independence referendum had turnout of 84.5%, reaching 90% in some areas, and even included 16 and 17-year-old voters for the first time, with around 75% participating. 97% of 16-17 years olds who voted said that they would vote again in future referendums. The EU referendum experienced 72% turnout, a record high for a UK-wide referendum, suggesting that the public does want the chance to have their say.
Yes – Historically, membership of parties has dropped as people have become less involved in formal politics
In 2015, only 1.0% of the electorate was a member of the Labour, Conservative or Liberal Democrat parties (compared with 3.8% in 1983). The Conservative Party had almost 3 million members in the 1950s, but this had dropped to around 130,000-150,000 by 2010. The 2015 Audit of Political Engagement found that only 30% of the public claimed to be a ‘strong supporter’ of a political party, of which only 76% said that they were certain to vote. This suggests a sharp drop in trust and engagement, leaving the parties with fewer volunteers, a limited talent pool to draw candidates from, and an increasing reliance upon wealthy donors and pressure groups. 
However… memberships of the Labour Party, many third parties, and pressure groups, is now growing
As of January 2018, Labour had 552,000 members. The SNP’s membership has grown from around 22,000 prior to the 2014 independence referendum, to over 120,000 by 2017. The RSPB has over a million members, including 195,000 youth members. This growth suggests that many people are politically engaged, and they are willing to participate when there are parties or groups that reflect their particular political views.