Final answer:
The European pursuit of an all-water route to Asia to avoid costly Silk Road taxes and their colonization attempts, despite the predominance of powerful Asian empires, was a characteristic of their influence in Southeast Asia.
Step-by-step explanation:
A characteristic of European trade and influence in Southeast Asia was the pursuit of new trade routes due to the costly and slow movement of goods on the Silk Road, which was heavily taxed by Muslim middlemen. Europeans sought an all-water, oceanic route to Asia to bypass these overland routes and gain direct access to spices, silk, and other valuable commodities. To that end, during the Pax Mongolica, European merchants like Marco Polo began travels as far as China, recounting tales of immense wealth and bringing back exotic goods which fueled the demand further. Such trade led to the spread of Buddhism and the influx of Islamic influence via the Silk Road. Despite establishing colonies like Batavia in Indonesia and occupying the Philippines, Europeans could not overpower the rich and formidable Asian empires.