J. E. Lighter, Random House Historical Dictionary of American Slang (1994) suggests that bash in the sense of "party" originated outside the United States, perhaps from the idiom "on the bash":
bash n. ... 2.a. a celebration or feast, esp. a boisterous party. [The term seems to have entered U.S. slang via the armed forces during WWII.]
[Examples:] 1901 in OEDS: Let us go and do a bash! ... [1924 in OEDS: The village tailor ... had an unfortunate weakness for getting terribly "on the bash" perhaps twice a year.] 1944 E.H. Hunt Limit 123: Quite a bash, wasn't it? 1944 AS XIX 310: Lieutenant Joe Klass, a young American Spitfire pilot, wrote late in May of the new colloquial vocabulary of American prisoners of war in German camps, [including] bash, banquet.
Eric Partridge, A Dictionary of Slang and Unconventional English, eighth edition (1984) partly corroborates that theory of origin, in this entry for bash:
bash, n. ... 4. A lively visit or experience or time: Can[adian]: since ca. 1955. Daily Colonist, 2 April 1967. 'The party of 48 chartered a bus for a night on the town including a bash at the Old Forge.' ... 8. In on the bash, on a drunken spree: C.20. Capt. R,W. Campbell, Private Spud Tamsen, 1916
But Partridge also notes that bash as a verb has long had a seemingly related slang meaning in Australian English:
bash; bash it; give it a bash. 'To indulge in a bout of heavy drinking' ([Sidney J.] B[aker, Australia Speaks], 1953): Aus[tralian]: since ca. 1935
On the other hand, John Ayto & John Simpson, The Oxford Dictionary of Modern Slang (1992), asserts that bash in the sense of "party" is of U.S. origin:
bash noun 1 on the bash: a Scottish and NZ On drinking bout. 1919–. ... 3 orig US A good time; a spree; a party. 1948–.
In general, American slang dictionaries Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) has this entry:
bash ... 2 n 1940s A party, esp a good, exciting one: Her little soiree turned into a real bash.
But Tom Dalzell, Flappers 2 Rappers: American Youth Slang (1996) asserts that bash had a narrower sense in the 1940s, as a verb:
bash To eat to excess
The relatively few slang dictionaries that address do in the sense of "party" suggest that it is a more formal affair than a bash is. From Jonathon Green, The Dictionary of Contemporary Slang (1984):
do n. 1. a party, a celebration, dinner, etc; often reasonably formal Pynchon;
Harold Wentworth & Stuart Flexner, Dictionary of American Slang, first edition (1960), traces the word (in U.S. usage) to 1952:
do ... n. A social affair or party. 1952: "I go to the Washington's Birthday dinner dance and a few of the main do's just to put in an appearance." Budd Schulberg, Holiday [magazine] Jan., 42/1.
Paul Beale, Partridge's Concise Dictionary of Slang and Unconventional English (1989) suggests that the term may be older and non-American, although the cited instance may be simply coincidental, given how basic the word do is:
do, n. An entertainment, a social function. In New Statesman and Nation, 23 Sep. 1933, we read of 'a famous West Indies cricketer, who speaks perfect English' (Constantine, no doubt) being puzzled by the phrase a slap-up do, applied to a tea. The puzzlement was admittedly caused more by the slap-up than by the do, though the juxtaposition may also have been partly the cause. In this sense, do obtained in dial. as early as 1820.
To summarize, bash in the sense of party may originate from kindred usage or phrases in the UK, Australia, Canada, the United States, or German POW camps; and do may be traced to the United States in 1952 or to Britain in 1820.
In general, a do seems more formal than a bash—and the phrase slap-up do doesn't tend in a contrary direction, as slap-up has virtually nothing in common with slap-dash. Partridge's Eighth Dictionary of Slang defines slap-up as meaning "Excellent; superior, first rate; grand."
If bash in the sense of party did indeed emerge from bash in the sense of "a drinking spree,"then perhaps the violence of the word bash is related to activities pursued during the spree or perhaps the state of the participant's head the next morning. This is purely speculative, however, as none of the references I consulted hazarded an explanation for the word choice.