Actually, a good modern analogy might be to look at how white mainstream America views the Black Lives Matter movement currently1: open hostility from social conservatives, and a lot of patronizing disagreement on methods from Liberals and Moderates.
Where the analogy (probably?) breaks down is that there were a great many conservatives who flat out wanted the man dead (as evidenced by later events), and that the FBI felt he was a danger to the State, and treated him accordingly.
I think your best view of the attitude he had to deal with from white liberals comes from his Letter from a Birmingham Jail, which was written to address their criticisms.2 From it, we can discern the following criticisms from "Moderates" who claimed to be supportive:
- He was bringing in non-residents for his actions.
- The demonstrations themselves were problematic and frivolous.
- Private negotiation should be used instead of public disruption.
- He was creating lots of "tension".
- He was attacking a relatively supportive government before it had a fair chance it help them itself.
- He was demanding change too quickly.
- He was breaking laws.
- He was inducing whites to violence.
- He was inciting passions in his own people that might be leading to violence on their part.
Here's a video of MLK on NBC's Meet the Press in 1965 receiving this exact same treatment from a panel of white journalists two years later. All these criticisms will sound quite fresh, if you follow the discussions around current American social justice movements. Surprisingly little seems to have changed in that regard.
I have almost reached the regrettable conclusion that the Negro's
great stumbling block in his stride toward freedom is not the White
Citizen's Counciler or the Ku Klux Klanner, but the white moderate,
who is more devoted to "order" than to justice; who prefers a negative
peace which is the absence of tension to a positive peace which is the
presence of justice; who constantly says: "I agree with you in the
goal you seek, but I cannot agree with your methods of direct action";
who paternalistically believes he can set the timetable for another
man's freedom; who lives by a mythical concept of time and who
constantly advises the Negro to wait for a "more convenient season."
1 - I am not equating the two directly. I'm just comparing the response they have received.
2 - Linked here is the letter that was the main catalyst for King's letter, written to attempt to undercut his support while he was conveniently in jail. As the last thing his jailors wanted was him responding, King's letter was written in the margins of newspapers and smuggled out piecemeal. However, its clear from the response that he was addressing more than just that one letter.