I usually balance an equation like this in acid and then convert to base at the end.
You can find the detailed procedure here.
1. Half-reactions
#"SO"_3^"2-" → "SO"_4^"2-"#
#"Cu"^"2+" → "Cu"^+#
2. Balance all atoms other than #"H"# and #"O"#
#color(red)(1)"SO"_3^"2-" → color(red)(1)"SO"_4^"2-"#
#color(blue)(1)"Cu"^"2+" → color(blue)(1)"Cu"^+#
3. Balance O
#color(red)(1)"SO"_3^"2-" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-"#
#color(blue)(1)"Cu"^"2+" → color(blue)(1)"Cu"^+#
4. Balance #"H"#
#color(red)(1)"SO"_3^"2-" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-" +color(olive)(2)"H"^+#
#color(blue)(1)"Cu"^"2+" → color(blue)(1)"Cu"^+#
5. Balance electrons
#color(red)(1)"SO"_3^"2-" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-" +color(olive)(2)"H"^+ + color(orange)(2)"e"^"-"#
#color(blue)(1)"Cu"^"2+" + color(orange)(1)e^"-" → color(blue)(1)"Cu"^+#
6. Equalize electrons transferred
#color(brown)(1) × [color(red)(1)"SO"_3^"2-" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-" +color(olive)(2)"H"^+ + color(orange)(2)"e"^"-"#
#color(brown)(2) × [color(blue)(1)"Cu"^"2+" + color(orange)(1)e^"-" → color(blue)(1)"Cu"^+]#
7. Add the two half-reactions
#color(brown)(1) × [color(red)(1)"SO"_3^"2-" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-" +color(olive)(2)"H"^+ color(red)(cancel(color(black)(color(orange)(2)"e"^"-"')))#
#color(brown)(2) × [color(blue)(2)"Cu"^"2+" + color(red)(cancel(color(black)(color(orange)(1)e^"-"))) → color(blue)(2)"Cu"^+]#
#stackrel(————————————————————)(color(red)(1)"SO"_3^"2-" + color(blue)(2)"Cu"^"2+" + color(olive)(1)"H"_2"O" → color(red)(1)"SO"_4^"2-" + color(blue)(2)"Cu"^+ +color(olive)(2)"H"^+)#
8. Convert to base
#color(red)(1)"SO"_3^"2-" + color(blue)(2)"Cu"^"2+" + color(red)(cancel(color(black)(color(olive)(1)"H"_2"O"))) → color(red)(1)"SO"_4^"2-" + color(blue)(2)"Cu"^+ +color(red)(cancel(color(black)(color(olive)(2)"H"^+)))#
#color(red)(cancel(color(black)(color(olive)(2)"H"^+))) + 2"OH"^"⁻" → stackrelcolor(blue)(1)(color(red)(cancel(color(black)(color(olive)(2)))))"H"_2"O"#
#stackrel(————————————————————)(color(red)(1)"SO"_3^"2-" + color(blue)(2)"Cu"^"2+" +2"OH"^"⁻" → color(red)(1)"SO"_4^"2-" + color(blue)(2)"Cu"^+ + color(olive)(1)"H"_2"O")#
The balanced equation is
#"SO"_3^"2-" + 2"Cu"^"2+" +2"OH"^"⁻" → "SO"_4^"2-" + 2"Cu"^+ + "H"_2"O"#
