Re "Has anyone ever [...] taught a course in higher algebra (in the ∞-categorical sense)?":

the following very new "resource" may be seen to spiritually qualify for this thread:

François Métayer: Homotopy theory of strict ω-categories and its connections with homology of monoids. Three-lecture course at the conference 'Categories in Homotopy Theory and Rewriting September 25 - 29, 2017' CIRM Marseille FRANCE

Needless to say, only spiritually (and arguably, not even that) since the central concept of the monograph 'Higher Algebra' are $(\infty,1)$-categories, which are what the OP is asking about, and which are quite different from the 'strict $\omega$-categories' in the sense of [R. Street: The algebra of oriented simplexes. J. Pure Appl. Algebr. 49 (1987) 283-335], which are what the CIRM course is about.

Abstract for the course on the CIRM website.

Abstract : In the first part, we describe the canonical model structure on the category of strict ω-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as ω-categories freely generated by polygraphs and introduce the key notion of polygraphic resolution. Finally, by considering a monoid as a particular ω-category, this polygraphic point of view will lead us to an alternative definition of monoid homology, which happens to coincide with the usual one.

MSC Codes : 18D05 - Double categories, 22-categories, bicategories, hypercategories 18G10 - Resolutions; derived functors 18G50 - Nonabelian homological algebra 18G55 - Homotopical algebra

References mentioned at the beginning of the course.

In Lecture 1, the following are mentioned by the lecturer as references for the course (maybe giving them here helps readers to decide whether to watch):

• D. G. Quillen: Homotopical Algebra (Lecture Notes in Mathematics)

• M. Hovey: Model categories Mathematical Surveys and Monographs 63, AMS, Providence, RI, 1999 (x + 209 pages).

• T. Beke: Sheafifiable homotopy model categories Math. Proc. Camb. Phil. Soc. vol.129 (2000), no.3, pp.447-475

• T. Beke: Sheafifiable homotopy model categories, Part II Journal of Pure and Applied Algebra vol.164 (2001), no.3, pp.307-324

Remarks.

• It seems preferable to use the course via the CIRM website instead of a large video hosting site, since CIRM site has additional structure, e.g.(so far only for Lecture 1) clickable buttons named by keywords, which take one to relevant points in the lectures. These are, in chronological order

Lecture 1

lifting properties : at 05:27

model structures : at 17:40

Smith's theorem : at 23:42

$\omega$-categories : at 32:24

weak equivalences : at 55:59

cylinder category : at 1:21:07

Lecture 2,3: sadly, the clickable buttons on the CIRM version cease after Lecture 1

• The course isn't primarily addressed at students, rather is a "mini-course" (this is the description on the conference's page integrated into a conference. It is therefore only an approximation to a graduate course. In particular, there aren't any exercises or tutorials.

Also, the above short course does not have the "goal to develop homological algebra from the "higher" point of view" (OP, 1st par.), yet it perhaps meets the rather permissive

"I'd like to know of any attempt at teaching anything related, "

requirement in the OP.