nt.number theory - Algorithm for the class field tower problem?

There's little if nothing to add to Cam's answer, except that I want to point out that there is a big technical difference between class field towers and $p$-class field towers.
I have never seen any conjecture in the direction of the statement "if $K$ has infinite class field tower, then some subfield of the class field tower has infinite $p$-class field tower for some prime $p$". All known infinite class field towers in fact come from some $p$-class field tower, for which Golod-Shafarevich applies.

Thus general class field towers are a very difficult topic. For $p$-class field towers, on the other hand, I would guess that most specialists indeed think that if such a tower is infinite, then some subfield satisfies the Golod-Shafarevich bound. In this connection, see

• F. Hajir, On the growth of $p$-class groups in $p$-class field towers,
  J. Algebra 188, No.1, 256-271 (1997)

But even if this were known, there would not be a terminating algorithm for deciding the finiteness of the $p$-class field tower. There are nontrivial cases in which the $2$-tower was shown to be finite; for some recent calculations see e.g.

• H. Nover, Computation of Galois groups associated to the 2-class towers of some imaginary quadratic fields with 2-class group $C_2 times C_2times C_2$,
  J. Number Theory 129, No. 1, 231-245 (2009)

This approach shows that certain types of class groups in small subfields prevent the $p$-class field tower from becoming infinite for group theoretic reasons. But there's a large gap between these results and Golod-Shafarevich, where no one really knows what is happening.