STAR will tolerate the '*' in the filenames command as this gets expanded by the command line. However, this may cause issues depending on what list it expands to. This argument should only take one sample which could be two files if paired. --readFilesIn /path/to/read1 [/path/to/read2]

Also, to my knowledge, STAR does not output to stdout like other aligners. Instead it writes directly to mapped_STAR/${base}Aligned.sortedByCoord.out.bam So you are piping irrelevant info into samtools sort which is overriding the output file.

Also, with the --outSAMtype BAM SortedByCoordinate option, you don't need samtools sort.

Here's a possible correction. I formatted according to my preference as well, but the important bit is to correct the input to outFileNamePrefix and to not override the bam output.

mkdir -p mapped_STAR
for i in trimmed_fastq2/*.trimmed.fastq
do
    base=$(basename ${i} .trimmed.fastq)
    STAR \
      --genomeDir genome_map/baboon_index \
      --runThreadN 16 \
      --readFilesIn ${i} \
      --outFileNamePrefix mapped_STAR/${base} \
      --outSAMtype BAM SortedByCoordinate \
      --outSAMunmapped None \
      --outFilterMismatchNmax 3 \
      --outFilterMultimapNmax 1 \
      --outSAMattributes All

        samtools index mapped_STAR/${base}Aligned.sortedByCoord.out.bam
done

if you have paired reads, these specific lines could change:

for i in trimmed_fastq2/*READ1.trimmed.fastq
base=$(basename ${i} READ1.trimmed.fastq)
--readFilesIn ${i} ${i/READ1/READ2} \

Final note, I don't think there is something fundamentally wrong with using echo and cut to get the basename, but it's not my preference because, for example, sometimes I might have a filename with the "." within the base part and not realize it...