The chemistry of volcanic rocks gives a lot of clues as to the origin of the melt. The athenosphere is actually a bit poorly defined in this sense (it's a mechanical classification, not a chemical classification). Regardless, it's easy to distinguish magmas with a pure mantle source from others. MORB (mid ocean ridge basalt) is a common acronym for volcanic rocks with a chemistry that indicates they're essentially pure mantle melt.
In the most common type of volcanoes you see on land (arc volcanoes), magma forms due to the introduction of water and other volatiles that lower the melting point of the upper mantle. That's what happens at arc volcanoes like the Cascades or Mt Fuji in Japan. It's not exactly coming from the athenosphere in that case, and it's often the lithospheric mantle and lower crust that are being melted. It's not heat that causes it, but instead the introduction of water.
Volcanoes can also form due to the introduction of extra heat, as happens above hotspot volcanoes (e.g. the Galapagos or Hawaii). In many of those cases, you're basically seeing the athenosphere supplying extra heat to the mantle lithosphere and crust and melting it. The magma usually isn't coming directly from the athenosphere in those cases (though some of it can and does). However, flood basalts are an extreme case of hotspots, and their chemical signatures suggest that it's more or less melt directly from the athenosphere.
Finally, one of the most common types of volcano on the planet does come directly from the athenosphere: mid ocean ridge systems. (Where the term MORB comes from) Those are almost entirely deep in the oceans, so non-geologists don't think about them as much, but they make up the bulk of volcanic activity on Earth. In those cases, you're essentially bringing the athenosphere up and melting more and more of it as it rises. Those are the cases where magma is most directly sourced from the athenosphere.