Step 1: Recall the purpose of cross-measure boreholes.
Cross-measure boreholes are drilled from underground roadways into surrounding strata above or below the coal seam.
Step 2: Identify their application.
These boreholes are primarily used to drain methane from adjacent strata before it enters mine workings.
Hence,
\[
\boxed{\text{Methane drainage}.}
\]
Therefore,
\[
\boxed{(A)}
\]
is the correct answer.