I stopped Integrate after quite a while -- there may be no closed form for this integral. However, NIntegrate will deliver a number, given values for s and b.
nint[s_, b_] :=
NIntegrate[
Exp[-(2 \[Pi] s Abs[x - y])/b]*Cos[(2 \[Pi] s)*(x - y)]*
Sin[1 \[Pi] x] Sin[1 \[Pi] y], {x, .001, 1}, {y, .001, 1}]
p1 = Plot3D[nint[s, b], {s, 0, 2}, {b, 0, 2}, AxesLabel -> {"S", "B"}]
