A little hard to quantify Hindu influences in the continental Southeast Asia (SEA) region but I can give a brief overview of where they came from.

During the time when ships relied on wind power and currents to get from place to place, Southeast Asia was basically in the middle of a trade superhighway. Generally, from October to April, the winds blew from China down to SEA and then west to India. When that happened, Chinese traders would sail to SEA and trade. From May to September, the winds blew from India to SEA and then northeast to China. The Chinese traders would return, and the Indian traders would arrive and conduct trade not just in SEA and Indian goods, but also in the Chinese goods the Chinese traders had sold to the SEAsians.

Because of this, SEA was heavily influenced by Indian and Chinese culture. One of those influences was religious in nature. We don't know how exactly it happened, whether it was a result of traders or missionaries or some other factor, or whether it was conversion from the ground up or from the top down, but Indian religions made a strong impact on the region.

Indian influence resulted in Hindu-Buddhist kingdoms spread throughout SEA. These were kingdoms that had Hinduism as the state religion, or Buddhism as the state religion, or some combination of the two. These existed as far back as 1st century CE (The Kingdom of Funan in Cambodia) and the last of them fell around 1500CE (the Majapahit Empire). You can see vestiges of this last empire in Bali, where Majapahit royalty fled when their capital fell, and which still practices a unique form of SEAsian Hinduism.

In the continental SEA region, the predominant influence was Hindu at first. The Kingdom of Funan was Hindu, Chen-La, which came after, was Hindu, and the Angkor (Khmer) Empire, which Suryavaman II once ruled over, was Hindu. We think the first major contact with Buddhism happened around 250BC, when Ashoka, a Buddhist emperor in India, was sending out missionaries. So as the Thais formed their own states and kingdoms, and then eventually formed the Kingdom of Thailand as we know it, there was plenty of time for Hinduism to influence their culture.

It's a little difficult to draw a line between Hinduism and Buddhism of the era in SEA. There's overlap between the deities, for example. Some Hindu influences that can still be seen in Thailand, where practically everyone identifies as a Buddhist:

Most Thais will be familiar with the Hindu epic, the Ramayana. There, it is called the Ramakien and is considered Thailand's national epic. In fact, massive statues of characters from it can be seen in Bangkok airport.

The national emblem of Thailand is the Garuda. The Garuda turns up in Buddhist mythology, but is also considered a mount of the Hindu deity, Vishnu. If you go to the Intercontinental Hotel in Bangkok, there's a shrine devoted to Vishnu, with Vishnu standing on the shoulders of Garuda, and it gets worshippers who wouldn't hesitate to call themselves Buddhists.

One of the most popular deities in Thai Buddhism is the Four-Faced Buddha, which is the Thai representation of the deity Brahma, the creator god with four faces. Worship of the Four-Faced Buddha can be seen most famously at Erawan Shrine in Bangkok.

I'm sure there are more examples but as I said, it's hard to quantify them without pointing out every single little detail. Suffice to say while most Thais identify as Buddhists, they don't see Hindu influence as "anti-Buddhist", or even as "Hindu". The two religions (from that culture and that era, anyway) are just not that distinct.

For more on Southeast Asian history, Southeast Asia in the Age of Commerce by Anthony Reid has a lot of good info.

Thailand: Into the Spirit World by Mariane Guelden is written for a popular audience, but it gives a good overview of modern Thai religious life. It also has several pictures that help in spotting the cross-religious influences.

Finally, Hinduism in Thai Life by Desai Santosh - does what it says on the tin.