ボクシングし始めたい→『ボクシングする』(verb) + 『始めたい』
ボクシングを始めたい→『ボクシング』(noun) +『を』+『始めたい』
Regarding this example, both mean the same, but the latter is more natural, I think.
For example, 料理し始めたい sounds you want to start cooking now, on the other hand, 料理を始めたい sounds you want to do cooking in near future so, are planning to cook at home or go to cooking school, something like that.