It is always prudent to screen proposed oligo sequence for potential off-target RNA interactions using BLAST. Partial homologies between a Morpholino's inverse complement and an RNA which are reported by BLAST are often in regions of the RNA where binding a Morpholino is unlikely to alter gene expression. If there is significant homology in a region where binding a Morpholino is likely to alter a transcript's expression, have us design another oligo.
Whether a BLAST hit presents a potential off-target effect depends on where the hit is located on an RNA as well as the degree of sequence similarity. Different people have different risk tolerance and you'll have to judge whether the risk is appropriate for your experiment. First BLAST the oligo's target sequence and consider the degree and location of similar sequence prior to ordering an oligo synthesis. Then you will also need to do subsequent empirical testing to check for off-target RNA interactions. Selecting and performing a specificity control experiment is an important step during your Morpholino experiments; specificity controls may include using a second non-overlapping oligo, doing an mRNA rescue, or using the oligo in a mutant that is null for the RNA the Morpholino targets.
BLAST the invert complement of the Morpholino sequence (the oligo's target sequence) in order to find places where the Morpholino might bind to off-target mRNA. For a translation blocker, at least BLAST the transcript sequences. For a splice modifier, at least BLAST genomic sequence. It is possible that a sequence from a 5'-UTR might occur elsewhere near a splice junction, or that a splice junction sequence might occur in a 5'-UTR, so for the most thorough check for unexpected interaction with RNA it is best to BLAST each oligo target against both transcript and genomic sequence.
Remember when searching for homologous targets that Morpholinos will only block translation when targeted to the UTR or first 25 bases of coding sequence. Also, Morpholinos can affect splicing if targeted in introns near intron-exon boundaries. If the Morpholino has homology to an off-target mRNA outside of these limited regions, binding of the oligo to the mRNA is less likely to affect expression of the off-target mRNA (though blocking regulatory sequences (e.g. exonic splice enhancers, 3'-UTR miRNA targets, etc.) may affect expression).
I usually suggest that the fraction of homologous bases should be below about 80%, but that cutoff % ignores important considerations about the distribution of the mispairs through the oligo. We have long stated that 14 to 15 contiguous bases of homology is the minimum inactivating length for a Morpholino (that is, less than a 14-mer oligo won't efficiently block translation), but if you flank 10 bases of homology with a mispair at either side and then add some runs of homologous bases at the outside borders, you can still get a knockdown. High CG content can make shorter homologous sequences active, since CGs pair with greater stability than ATs; in general, homologous sequences with high predicted Tm should be avoided if they are in an area which might affect gene expression. Recent reports suggest that homologies with as little as 11 bases matched can have off-target effects (see: BLAST homology and specificity controls), though the 11-base sequence reported has an unusually high Tm (see: Paper on Morpholino specificity and CRISPR-MO combination). Five mispairs spread throughout a 25-mer almost always gives loss-of-knockdown (so we sell five-mispair oligos as specificity controls, though there are better specificity controls). If you place all five mispairs in one end of the oligo, you still get 20 contiguous complementary bases in a 25-mer and those 20 bases would retain considerable antisense activity. My point is that when you find a partially-homologous region, following a rule of thumb like "less than 80% homology is OK" can lead to trouble -- you still need to look at distribution of the mispairs.
Watch for multiple regions of homology, for example 12 bases of homology, 1 mismatch, and 12 more bases of homology. If you are just looking for contiguous homology you would interpret this as 12 contiguous bases of homology, but it is really a single-mismatch 25-base target that would very likely lead to strong knockdown by the Morpholino.
Here are a few more factors to consider: losing a C-G pair impacts the oligo activity more than losing an A-T pair (three H-bonds for G-C compared to two for A-T) and watch for forming G-T pairs, which are non-Watson-Crick pairing but still form two hydrogen bonds.
The BLAST algorithm is not an exhaustive sequence search, so it is not surprising you see some differences between different BLASTs of the same sequence. Screening using several databases is best and even with the same genome build, they will often produce different results; it really depends how much of your time you make available for the screen. BLASTING a sequence can sometimes catch a problem, but it is an endless rabbit-hole to explore so you have to set a limit on how long you will search a given sequence. In the end, a good physical specificity control should tell you if the targeting oligo is working specifically. BLASTing just helps eliminate some oligos up-front that will not survive the specificity control challenge.