Why does Brady's reagent include sulfuric acid and methanol?
2,4-DNPH is usually dissolved in an alcoholic solvent, together with sulfuric acid, when used as a reagent to detect aldehydes and ketones (Brady's reagent). The sulfuric acid is a source of $\ce{H+}$, which catalyses the reaction between the hydrazine group and the carbonyl compound, as depicted in the mechanism below:



The final hydrazone product is the precipitate which signals a positive test.

The solvent used is typically methanol or ethanol. You are correct that these could theoretically react with sulfuric acid: dimethyl ether is the most likely product. However, the concentration of sulfuric acid is typically not sufficiently high, and these side reactions do not take place.