How much does the Bodhisattva way influence Theravadan practices?

Compassion and the Bodhisatta path are two different things; compassion is a quality of mind, the Bodhisatta path is a determination. The latter is based, one could say, on a high level of compassion, but it is not equivalent to compassion. It is the determination to not enter into freedom from suffering until one is able to (in Theravada Buddhism) become a fully-enlightened Buddha, one who can teach others without having been taught oneself. From the Jataka Commentary:

62. Then loosened I my matted hair,
And, spreading out upon the mud
My dress of bark and cloak of skin,
I laid me down upon my face.

63. "Let now on me The Buddha tread,
With the disciples of his train;
Can I but keep him from the mire,
To me great merit shall accrue."

64. While thus I lay upon the ground,
Arose within me many thoughts:
"To-day, if such were my desire,
I my corruptions might consume.

65. "But why thus in an unknown guise
Should I the Doctrine's fruit secure?
Omniscience first will I achieve,
And be a Buddha in the world.

66. "Or why should I, a valorous man,
The ocean seek to cross alone?
Omniscience first will I achieve,
And men and gods convey across.

67. "Since now I make this earnest wish,
In presence of this Best of Men,
Omniscience sometime I'll achieve,
And multitudes convey across.

68. "I'll rebirth's circling stream arrest,
Destroy existence's three modes;
I'll climb the sides of Doctrine's ship,
And men and gods convey across.

http://www.sacred-texts.com/bud/bits/bits001.htm

Compassion, on the other hand, is just a mind state that can be cultivated by anyone; we say a Buddha has great compassion because of his sacrifice, but anyone can cultivate it as a meditation practice. See Wikipedia on Karuṇā.

The reason why karuṇā is considered a side practice is because it cannot itself lead to enlightenment, taking as it does concepts (living beings) as its object.

Now, zeal consisting in desire to act is the beginning of all these things. Suppression of the hindrances, etc., is the middle. Absorption is the end. Their object is a single living being or many living beings, as a mental object consisting in a concept.

Path of Purification, IX.102

So the best it can lead to directly is worldly absorption (lokiya samatha jhana).

Since the goal of Theravada Buddhism is enlightenment here and now, meditation based on concepts takes a back seat role to meditation on the five aggregates, four foundations of mindfulness, etc., which take ul

karuṇā can play an important role in quelling cruelty in the mind, which is a hindrance on the path, just as metta can play an important role in quelling anger. They are useful practices, just not enough to lead to understanding of reality.

As for the Bodhisatta path itself, Theravada Buddhists in general tend to favour not reinventing the wheel; the path has been opened, why not walk it and encourage others to walk it as well? The only way the path will stay open is if people continue to practice and follow it to its goal; once everyone decides its better to wait until one can become a fully enlightened Buddha, the path will be effectively closed; the question then arises, what was the point of becoming a fully enlightened Buddha in the first place?

The Mahayana, if I'm not mistaken, gets around this quandary of everyone wanting to lead and no one to follow by suggesting that the arahant is not actually enlightened and still has further to go. Or something like that.

So, while there are many notable examples of Bodhisattas in Theravada Buddhism (the king of Thailand, I think, may have hinted at it), for the most part the idea is to keep the path that is already open open long enough to get as many people across to safety as possible. This seems preferable to waiting countless more eons and letting those people undergo untold suffering before the path can be opened again (assuming one is successful in attaining Buddhahood at all).