strand-aware RNA-seq data
There is no such thing. There are strand-specific library preparations, though, which generate strand-specific (relative to the mRNA) reads, which is probably what you mean.
Does STAR perform strand-aware mapping, which we can use downstream to get antisense transcripts
You can't get transcripts, sense or antisense - you can get mappings and counts from STAR. What are the downstream analyses you intend to perform?
STAR maps reads to an indexed (most commonly annotated) reference genome, and it doesn't care about transcript strandedness. From these mappings, it can output many different things, which can be used for several downstream analyses. The three most commonly used outputs from STAR are:
read mappings to the genome, in BAM format. By default, strandedness has no importance: STAR maps the reads to the best location, regardless of strand, gene location or whatever. However, the parameter --outSAMstrandField intronMotif - which should be used only if you want to analyse downstream with Cufflinks, and have unstranded RNAseq dara - alters the default behaviour and adds XS strand attribute. Read the section Compatibility with Cufflinks/Cuffdiff from STAR's manual. If you have stranded data, you don't have to change anything in STAR.
read mappings to the transcriptome, in BAM format. Those are to be used with RSEM, mostly.
counts of reads mapping to genes, similar to HTSeq or featureCounts. Here STAR outputs three columns of counts, one for unstranded, one for stranded, and one for reverse stranded.
Most of these points were already clearly made by Alex Dobin in the thread Are any additional parameters for STAR to hand the stranded RNA-Seq data?, and the STAR manual.