After Norman Steenrod's famous paper, algebraic topologists routinely work in the framework of a "convenient category of topological spaces" which notably includes the requirement of being cartesian closed. Part of the reason is the need for fundamental constructions like path spaces and loop spaces. (The path space of a space $X$ in such a convenient category being an exponential $X^I$.) The convenience and need for cartesian closure makes itself strongly felt in basic constructions of operads, such as the linear isometries operad. See for example even early works of J.P. May, such as The Geometry of Iterated Loop Spaces, and $E_\infty$ Rings and $E_\infty$ Spectra (available at his web page).
Another extremely convenient consequence of cartesian closure is that it ensures that products interact well with colimit constructions (precisely, that the functor $X \times -$ preserves colimits). Not having this is a major inconvenience. For example, in order to have the basic result that the geometric realization functor (from simplicial sets to spaces) preserves products, it is necessary to have this nice interaction. Thus, the geometric realization functor does not behave well (e.g. does not preserve products) if it is taken to land in the category of topological spaces, but if one uses a convenient category such as compactly generated spaces, one obtains the desired result. You can find a discussion of these issues in the nLab.
Another application of this desired interaction is more theoretical: that for a convenient category of spaces $C$, the category of $T$-algebras $C^T$ for a Lawvere theory $T$ is monadic over $C$.
Perhaps the best way to say it is that lack of cartesian closure tends to make life very annoying for the purposes of algebraic topologists. A main purpose is to simplify arguments, and remove the hassle of not having products interact appropriately with quotient space constructions (and other delicate point-set topological considerations).