Balanced triplet brackets

JavaScript (ES6), 77 58 57 56 bytes

f=s=>s==(s=s.replace(/\(\+\)|\[:]|{\|}|<->/,''))?!s:f(s)

Test cases

f=s=>s==(s=s.replace(/\(\+\)|\[:]|{\|}|<->/,''))?!s:f(s)

console.log("Testing truthy values");
console.log(f("(+)(+)(+)"));
console.log(f("[[[:]:]:(+(+))]{<->|<(+)->[:]}(+)"));
console.log(f("<<<<<<<<<<<<->->->->->->->->->->->->"));
console.log(f("{|(+[:<-{|(+[:<->])}>])}"));
console.log(f("[[[:]:[:]]:[[:]:[:]]]"));
console.log(f("{[:](+)|<->{|}}[(+)<->:{|}(+)]"));

console.log("Testing falsy values");
console.log(f(":["));
console.log(f("<|>"));
console.log(f("(+(+)"));
console.log(f("[:][:](+[[:]):]"));
console.log(f("{|{|{|(+{|{|{|}}}}}+)}[:]{|}"));
console.log(f("{{||}}"));
console.log(f("<<->-<->-<->>"));
console.log(f("[(+):((+)+)+(+(+))]"));
console.log(f("<<<<<->->->->->->"));

sed, 28 27 bytes

:
s#(+)\|\[:]\|{|}\|<->##
t

sed doesn't have a concept of truthy/falsy, so I'm considering an empty string truthy and a non-empty string falsy. This checks out if we consider the conditional /^$/.

Thanks to @Neil for golfing off 1 byte!

Try it online!

Python, 77 bytes

lambda s:eval("s"+".replace('%s','')"*4%('(+)','[:]','{|}','<->')*len(s))==''

Uses Arnauld's replacement idea. Generates and evaluates a long string like

s.replace('(+)','').replace('[:]','').replace('{|}','').replace('<->','').replace('(+)','').replace('[:]','').replace('{|}','').replace('<->','')

to cycle between replacing all the bracket types. Then, checks if the result is the empty string.