Apart from the question whether arm swing is actively controlled or merely passive, there seems to be no consensus on why humans swing their arms during gait (i.e. what the purpose of arm swing is, if it has a purpose at all). Suggested reasons include minimizing energy consumption, optimizing stability, and optimizing neural performance.

Source :Gait and Posture, Volume 38, Issue 4

In general, humans tend to swing their arms in sync (but 180° out-of-phase) with their walking strides. Although the muscle-driven arm swinging must incur at least some energy costs, three recent experimental studies have shown that not swinging actually increases the overall energy consumed by walking.

Several hypotheses have been proposed to explain arm-swing, and its energy cost/benefits, but as yet there is no general agreement.

Independent of how arm swing is executed, it appears to play an important part during human locomotion. However, what this role is exactly, is still unknown. Several hypotheses have been formulated, among which:
(a) reducing vertical displacement of the center of mass (COM) (Hinrichs, 1990; Murray et al., 1967; Pontzer et al., 2009;
(b) reducing angular momentum around the longitudinal axis (Bruijn et al., 2008; Bruijn et al., 2011; Collins et al., 2009a; Elftman, 1939; Hinrichs, 1990; Park, 2008);
(c ) reducing angular movement around the longitudinal axis (Fernandez Ballesteros et al., 1965; Murray et al., 1967; Pontzer et al., 2009);
(d) reducing the ground reaction moment (GRM) (Collins et al., 2009a; Li et al., 2001; Witte et al., 1991);
(e) increasing (local) stability (Ortega et al., 2008) / balance recovery after perturbations (Bruijn et al., 2010; Hof, 2007; Marigold et al., 2002; Pijnappels et al., 2010);
(f) facilitating leg movement (Meyns et al., 2013) and;
(g) minimizing energetic costs (Collins et al., 2009a; Ortega et al., 2008; Umberger, 2008).
These hypotheses cannot be seen entirely separate from each other, and are in some cases even entirely interdependent."

Source : Influence of arm swing on cost of transport during walking Biology Open 2019, May 2019.